On Central-Peripheral Appendage Numbers of Uniform Central Graphs

TL;DR

This paper introduces the central-peripheral appendage number for uniform central graphs, providing formulas based on graph parameters and establishing bounds, including a maximum of 6 when the diameter exceeds 2.

Contribution

It defines and computes the central-peripheral appendage number for UCGs, offering new formulas and structure theorems based on graph properties.

Findings

A_{ucg}(C, P) is computed using radius and diameter of P and the completeness of C.

A_{ucg}(C, P) ; 6 when diam(P) > 2.

Structural characterizations of UCGs are provided.

Abstract

In a uniform central graph (UCG) the eccentric verticies of a central vertex is the same for all central verticies. This collection of eccentric verticies is the centered periphery. For a pair of graphs $(C, P)$ the central-peripheral appendage number, $A_{ucg}(C, P)$, is the minimum number verticies needed to be adjoined to the graphs $C$ and $P$ in order to construct a uniform central graph H with center C and centered-periphery P. We compute $A_{ucg}(C, P)$ in terms of the radius and diameter of P and whether or not $C$ is a complete graph. In the process we show $A_{ucg}(C, P)\leq 6$ if $diam(P) > 2$. We also provide structure theorems for UCGs in terms of the centered periphery.

Equations

$$\mathcal{C}\mathcal{P}(G)=\bigcup _{z\in \mathcal{Z}(G)}EC(z),$$

$$\bullet$$

$$P_i^{\prime }\not\subset \bigcup _{j\ne i}{P}_j^{\prime }.$$

$$P_i\subset \bigcup _{j\ne i}{P}_j.$$

$$D_m=\{u\in V(G):d(u,\mathcal{Z}(H))=m\}$$

$$d(x_i,p)\le d(x_i,p^{\prime })+d(p^{\prime },p)=(r(H)-1)+1=r(H)$$

$$D_m^{\prime }=D_m\cap \bigcup _{i=1}^{k^{\prime }}V({\mathcal{L}}_i)\text{ and }{D}_m^{\prime \prime }(\overline{P})=D_m\cap \bigcup _{i=1}^{k^{\prime \prime }}V({\mathcal{L}}_i).$$

$$|D_1^{\prime \prime }|=|F_1|=|F_m|\le |D_m^{\prime \prime }|.$$

$$\kappa \le k^{\prime \prime }=|D_1^{\prime \prime }|\le |D_m^{\prime \prime }|\le |D_m|.$$

$$\mathcal{I}(H)=\bigcup _{m=1}^{r-1}{D}_m$$

$$V(G)=V(C)\cup V(P)\cup \{x_{i,j}:0\le i\le k,1\le j\le \rho \}$$

$$C$$

$$\kappa \le k=|D_1|=|\mathcal{I}(H)|=\kappa$$

$$G=\mathcal{G}(C,P,\overline{P},2)-\{x_{0,1},x_{0,2}\}.$$

$$4\le d(u,v)\le d(u,p)+d(p,v)\le d(u,p)+2.$$

$$G=\mathcal{G}(C,P,\overline{P},2)-\{x_{0,1},x_{0,2}\}$$

$$2\kappa \le 2k^{\prime \prime }\le |D_1^{\prime \prime }|+|D_2^{\prime \prime }|\le |\mathcal{I}(H)|=2\kappa ,$$

$$G=\mathcal{G}(C,P,\overline{P},2)-\{x_{0,1},x_{0,2}\}.$$

$$2\kappa \le 2k^{\prime \prime }\le |D_1^{\prime \prime }|+|D_2^{\prime \prime }|\le |\mathcal{I}(H)|=2\kappa +1$$

$$\kappa =|D_1^{\prime \prime }|\le |D_1|\text{ and }\kappa =|D_1^{\prime \prime }|\le |D_2^{\prime \prime }|\le |D_2|,$$

$$\kappa =|D_1^{\prime \prime }|\le |D_2^{\prime \prime }|\le |D_2|=\kappa .$$

$$I=\{i:x_k\text{ is adjacent to }{y}_i\text{ for some }{y}_i\in D_2\}.$$

$$P_k=\bigcup _{i\in I}{P}_i.$$

$$d(p,x_k)\le d(p,p_j)+d(p_j,x_k)\le 3$$

$$d(y_i,x_j)\le d(y_i,c)+d(c,x_j)=3,$$

$$G=\mathcal{G}(C,P,\overline{P},2)-\{x_{0,2}\}.$$

$$V(G)=V(C)\cup V(P)\cup \{x_i,y_j:1\le i\le k,0\le j\le k\}$$

$$C$$

$$\overline{Q}=\overline{P}\cup \{Q_0,Q_1\}-\{P_{\iota }\}.$$

$$Q_i=\{p\in V(P):p\text{ is adjacent to }{y}_i\}.$$

Full text

1 Introduction

Let $G=(V(G),E(G))$ be a simple graph. The eccentricity of a vertex $v$, denoted by $e(v)$, is defined by $max\{d(u,v):u\in V(G)\}$ where $d(u,v)$ is the distance between two vertices $u$ and $v$. A vertex $x$ is called an eccentric vertex of $v$ if $d(v,x)=e(v)$; and the set of all eccentric vertices of $v$ is denoted by $EC(v)$. The radius of $G$, $r(G)$ or $r$, is the minimum eccentricity of the vertices in $G$ and the diameter of $G$, $\text{diam}(G)$, is the maximum eccentricity of the vertices in $G$. A $(u,v)$-path is called a distance measuring path, or a dmpath, when the length of the path is $d(u,v)$. For a central vertex $u$, a $(u,v)$-dmpath is called a radial path if the length of the path is $r(G)$. For any nonempty subset $S$ of vertices in $G$, $\langle S\rangle$ represents the induced subgraph of $G$ by $S$. For terminology not defined in this paper, the reader is referred to [5].

Let $\mathcal{Z}(G)$ be the center of the graph $G$ and define the centered periphery of $G$ as

[TABLE]

the set of vertices that are far from the center. The periphery, which is the set of maximally eccentric vertices, coincides with centered periphery for some graphs. However, in many cases they differ. For instance, the graph in figure 1 has periphery $\{p_{0},p_{1},p_{2},p_{5},p_{6},p_{7}\}$ while the centered periphery is $\{p_{1},\ldots,p_{6}\}$.

A graph $G$ is called a uniform central graph, or UCG, if $EC(c)$ is same for all central vertices $c$ of $G$, i.e. $EC(c)=\mathcal{CP}(G)$ for every vertex $c$ in the center. It is known that a radial path in a uniform central graph contains only one central vertex [2]. In the study of UCGs it is also useful to define the set of “intermediate” vertices of the graph as $\mathcal{I}(G)=V(G)-(\mathcal{Z}(G)\cup\mathcal{CP}(G))$.

An appendage number of a graph $G$ is the minimum number of vertices to be added to $G$ to obtain a supergraph $H$ of $G$ so that $H$ satisfies some prescribed properties. Buckley, Miller and Slater [1] studied the appendage number of a graph $G$ satisfying that $G$ is the center of its supergraph. This result is extended by Gu [3], where the supergraph is a UCG with a given $G$ as its center. More recently S. Klavžar, K. Narayankar, and S. Lokesh [4] looked at appendage numbers of graphs where the supergraph is a UCG with $G$ as any subgraph.

In this paper, we study the appendage number for a pair of given graphs $(C,P)$, where the supergraph $H$ is a uniform central graph satisfying $\langle\mathcal{Z}(H)\rangle=C$ and $\langle\mathcal{CP}(H)\rangle=P$. We denote the central-peripheral appendage number, $A_{ucg}(C,P)$, as the minimum number of “intermediate” vertices needed to construct such a uniform central graph $H$. By convention $A_{ucg}(C,P)=\infty$ if there is no graph satisfying the above conditions.

Gu’s results in [3] depend on whether or not the desired center is a complete graph. This distinction also appears in our work. In light of this we organize our paper in the following way. Section 2 focuses on results that do not depend on the center. These results are then applied to graphs with specific centers; complete graphs in section 3 and all other graphs in section 4 to classify central-peripheral appendage numbers. The results in sections 3 and 4 are given in terms of sizes of various coverings of $P$. In section 5 computes the sizes of these coverings. Section 6 summarizes the previous work and describes the central-peripheral appendage numbers in terms of the diameter and radius of $P$. Finally, in section 7, using different techniques than Gu, we obtain her results as a corollary of our results.

2 General Centers

This section develops results about the structure of a uniform central graph $H$ in terms of $P=\langle\mathcal{CP}(H)\rangle$. We do this by studying coverings of the centered periphery.

A covering of a graph $G=(V,E)$ is a set $\overline{V}=\{V_{1},\ldots,V_{k}\}$ where $V_{i}\subset V$ with $\cup V_{i}=V$. We say $k$ is the size of the covering. A subcovering of a covering $\overline{V}$ is a subset of $\overline{V}$ that is also a covering.

Throughout this paper we are interested in coverings satisfying various properties. The simplest and most important of these cinditions is condition A. Let $P$ be a graph with covering $\overline{P}=\{P_{1},\ldots,P_{k}\}$. We say the pair $(P,\overline{P})$ satisfies

Condition A:

if for each $1\leq i\leq k$, there is a vertex $p\notin P_{i}$ satisfying $d(P_{i},p)\geq 2$.

When there is no confusion to the graph $P$, we simply say $\overline{P}$ satisfies condition A.

Elements of a cover can overlap. It is useful to minimize this overlap in some sense.

Lemma 2.1**.**

Let $\{P_{1},\ldots,P_{k}\}$ be a covering of $P$ satisfying condition A. Then there is a subcovering $\{P_{1}^{\prime},\ldots,P_{\eta}^{\prime}\}$ satisfying condition A and for each $i$ ($1\leq i\leq\eta)$

[TABLE]

That is, for each $i$ there exists $\tilde{p}_{i}\in P_{i}$ such that $\tilde{p}_{i}\not\in P_{j}$ for all $j\neq i$.

Proof.

Suppose there exists an $i$ satisfying

[TABLE]

Then $\{P_{j}:j\neq i\}$ is a subcovering satisfying condition A. ∎

The main idea of this paper is to take the question of determining $A_{ucg}(C,P)$ and to transfer it to questions about the size of coverings of $P$ satisfying various conditions. This is possible because for any UCG there exists a natural covering on $P$. It turns out this covering satisfying condition A. These coverings are used to obtain lower and upper bounds on $A_{ucg}(C,P)$. To do this we first introduce some notation.

For a uniform central graph $H$ let

[TABLE]

for $0\leq m\leq r=r(H)$. This gives a stratification of $H$ with $D_{0}=\mathcal{Z}(H)$ and $D_{r}=\mathcal{CP}(H)$. For an enumeration $D_{1}=\{x_{1},\ldots,x_{k}\}$ let $\mathcal{L}_{i}$ be the set of all radial paths from $C$ to $P$ containing $x_{i}$

Lemma 2.2**.**

Let $H$ be a uniform central graph with $\langle\mathcal{CP}(H)\rangle=P$. Then the graph $H$ induces a covering on $P$ satisfying condition A.

Proof.

Let $H$ be a uniform central graph with $\langle\mathcal{CP}(H)\rangle=P$. Furthermore, let $C=\langle\mathcal{Z}(H)\rangle$ and $D_{1}=\{x_{1},\ldots,x_{k}\}$ for some $k$. Let $P_{i}=V(P)\cap V(\mathcal{L}_{i})$. Then $\{P_{1},\ldots,P_{k}\}$ is a covering of $P$ since each radial path from $C$ to $P$ contains a vertex in $D_{1}$.

Assume $\{P_{1},\ldots,P_{k}\}$ does not satisfy the condition A. Then there is an $i$ satisfying that for every vertex $p\not\in P_{i}$ there is a $p^{\prime}\in P_{i}$ with $d(p,p^{\prime})\leq 1$. This shows $x_{i}\in V(C)$, a contradiction.

Since $x_{i}\in D_{1}$, there is a central vertex $c$ with $d(x_{i},c)=1$. For each vertex $v$ of $H$ not in $P$, $d(c,v)\leq r(H)-1$ and hence $d(x_{i},v)\leq r(H)$. If $p$ is a vertex in $P_{i}$, $d(x_{i},p)=r(G)-1$ from the construction of $P_{i}$.

If $p$ is a vertex in $P$ but not in $P_{i}$. Then there is a vertex $p^{\prime}\in P_{i}$ with $d(p,p^{\prime})=1$ and so

[TABLE]

and so $x_{i}\in V(C)$, a contradiction. ∎

Corollary 2.3**.**

If $H$ is a UCG with $P=\langle\mathcal{CP}(H)\rangle$, then $r(P)>1$.

Proof.

If $r(P)=1$, then there is $q\in V(P)$ with $e(q)=1$. Let $\overline{P}=\{P_{1},\ldots,P_{k}\}$ be the induced cover of $P$. Without loss of generality we may assume $q\in P_{1}$. Then for all $p\in V(P)$, $d(p,P_{1})\leq 1$ and condition A is not met. Hence by lemma 2.2 $P$ is not a centered periphery for any UCG. ∎

Corollary 2.4**.**

If $P$ is a graph with $r(P)\leq 1$, then $A_{ucg}(C,P)=\infty$ for all graphs $C$.

For $H$, a UCG with $C=\langle\mathcal{Z}(H)\rangle$ and $P=\langle\mathcal{CP}(H)\rangle$, we use the induced covering from lemma 2.2 to gain a better understanding of the structure of $\mathcal{I}(H)$.

Let $D_{1}=\{x_{1},\ldots,x_{k}\}$ and $\{P_{1},\ldots,P_{k}\}$ be the induced covering of $P$ defined as in lemma 2.2. Without loss of generality assume $\{P_{1},\ldots,P_{k^{\prime}}\}$ with $k^{\prime}\leq k$, is subcovering with non-empty elements . Also let $\overline{P}=\{P_{1},\ldots,P_{k^{\prime\prime}}\}$ and $\{\tilde{p}_{1},\ldots,\tilde{p}_{k^{\prime\prime}}\}$, with $k^{\prime\prime}\leq k^{\prime}$, be as in lemma 2.1. For $1\leq m\leq r(H)-1$ we define the following subsets of $D_{m}$.

[TABLE]

$D_{m}$ and $D_{m}^{\prime}$ are well-defined with respect to $H$, however $D_{m}^{\prime\prime}(\overline{P})$ depends on a choice of subcover. When this choice is clear from context, we simply use $D^{\prime\prime}_{m}$.

The following proposition describes the structure of $\mathcal{I}(H)$ that is the key to the rest of this paper.

Proposition 2.5**.**

Let $H$ be a UCG with $C=\langle\mathcal{Z}(H)\rangle$, $P=\langle\mathcal{CP}(H)\rangle$, $\overline{P}=\{P_{1},\ldots,P_{k^{\prime\prime}}\}$ and $\{\tilde{p}_{1},\ldots,\tilde{p}_{k^{\prime\prime}}\}$ as in lemma 2.1 with respect to the induced covering from lemma 2.2. Then any $(x_{i},\tilde{p}_{i})$-dmpath and $(x_{j},\tilde{p}_{j})$-dmpath are disjoint if $i\neq j$.

Proof.

For $L_{i}$ a $(x_{i},\tilde{p}_{i})$-dmpath and $L_{j}$ a $(x_{j},\tilde{p}_{j})$-dmpath with $i\neq j$, assume there exists a $y\in V(L_{i})\cap V(L_{j})$. Let $L_{i}^{\prime}$ be the subpath of $L_{i}$ from $x_{i}$ to $y$ and $L_{j}^{\prime}$ the subpath of $L_{j}$ from $y$ to $\tilde{p}_{j}$. The concatenation of $L_{i}^{\prime}$ and $L_{j}^{\prime}$ yields a $(x_{i},\tilde{p}_{j})$-dmpath and so $\tilde{p}_{j}\in P_{i}$, a contradiction. ∎

Corollary 2.6**.**

With the assumptions of proposition 2.5, $|D_{1}^{\prime\prime}|\leq|D_{m}^{\prime\prime}|$ for $1\leq m\leq r(H)-1$.

Proof.

For each $1\leq i\leq k^{\prime\prime}$ there exists $L_{i}$, a $(x_{i},\tilde{p}_{i})$-dmpath. Let $\mathcal{L}=\{L_{1},\ldots,L_{k^{\prime\prime}}\}$ and $F_{m}=V(\mathcal{L})\cap D_{m}^{\prime\prime}$. By proposition 2.5 $|F_{1}|=|F_{m}|$. Then by the construction of $F_{m}$

[TABLE]

∎

We often work with UCGs with few intermediate verticies. For such UCGs we can say more about thier structure.

Lemma 2.7**.**

Let $H$ be a UCG with $P=\langle\mathcal{CP}(H)\rangle$ and $r=r(H)$. Furthermore, let $\overline{P}=\{P_{1},\ldots,P_{k^{\prime\prime}}\}$ be a subcovering of the induced covering and $\{\tilde{p}_{1},\ldots,\tilde{p}_{k^{\prime\prime}}\}$ an associated set of vertices as in lemma 2.1. If $|D_{1}^{\prime\prime}|=|D_{r-1}^{\prime\prime}|$ then for each $P_{i}\in\overline{P}$ there is a unique $y_{i}\in D_{r-1}^{\prime\prime}$ such that $y_{i}$ is adjacent to every vertex in $P_{i}$. Furthermore, if $i\neq j$, then $y_{i}\neq y_{j}$.

Proof.

Let $D_{1}^{\prime\prime}=\{x_{1},\ldots,x_{k^{\prime\prime}}\}$. For each $i$, let $L_{i}$ be an $(x_{i},\tilde{p}_{i})$-dmpath and let $y_{i}$ be the vertex on $L_{i}$ adjacent to $\tilde{p}_{i}$. Then $y_{i}\in D_{r-1}^{\prime\prime}$. By proposition 2.5 $y_{i}\neq y_{j}$ when $i\neq j$. Furthermore $D_{r-1}^{\prime\prime}=\{y_{1},\ldots,y_{k^{\prime\prime}}\}$. We claim $y_{i}$ is adjacent to every vertex in $P_{i}$.

For a vertex $p\in P_{i}$ there is a $c\in\mathcal{Z}(H)$ and a $(c,p)$-radial path $L$ that contains $x_{i}$. Let $y$ be the vertex on $L$ adjacent to $p$. Since $y\in D_{r-1}^{\prime\prime}$, $y=y_{j}$ for some $1\leq j\leq k^{\prime\prime}$. Since $y_{j}$ is adjacent to $\tilde{p}_{j}$, there is a $(x_{i},\tilde{p}_{j})$-dmpath of length $r-1$. Therefore, $\tilde{p}_{j}\in P_{i}$ which means $j=i$, $y=y_{i}$ and $y_{i}$ is adjacent to $p$. ∎

Lemma 2.8**.**

Let $H$ be a UCG with $C=\langle\mathcal{Z}(H)\rangle$ and $P=\langle\mathcal{CP}(H)\rangle$ and assume the induced covering $\overline{P}=\{P_{1},\ldots,P_{k}\}$ satisfies the conditions of the subcovering in lemma 2.1. That is $D_{1}=D_{1}^{\prime\prime}$. Then each vertex in $C$ is adjacent to each vertex in $D_{1}$.

Proof.

Let $D_{1}=\{x_{1},\ldots,x_{k}\}$ and $\{\tilde{p}_{1},\ldots,\tilde{p}_{k}\}$ be an associated set of vertices to $\overline{P}$ as in lemma 2.1. For each vertex $c$ in the center and each $\tilde{p}_{i}$, there is a $(c,\tilde{p}_{i})$-radial path. By construction of $\overline{P}$ and definition of $\tilde{p}_{i}$, each $(c,\tilde{p}_{i})$-radial paths must contain $x_{i}$. Therefore, $c$ and $x_{i}$ are adjacent. ∎

For a graph $P$, let $\text{cov}_{A}(P)$ be the smallest size of a covering of $P$ satisfying condition A. Observe, from the definition of condition A it follows that $\text{cov}_{A}(P)\geq 2$ for any graph $P$ with $r(P)>1$. The following result is crirtical in obtaining lower bounds on $A_{ucg}(C,P)$.

Proposition 2.9**.**

If $H$ is a uniform central graph with $C=\langle\mathcal{Z}(H)\rangle$, $\langle\mathcal{CP}(H)\rangle=P$, $r=r(H)$ and $\kappa=\text{cov}_{A}(P)$, then $|\mathcal{I}(H)|\geq\kappa(r-1)$.

Proof.

Let $\{P_{1},\ldots,P_{k^{\prime\prime}}\}$ be a subcover of the induced cover on $P$ from $H$ from lemmas 2.2 and 2.1. By definition of $\kappa$ and corollary 2.6 it follows that

[TABLE]

Since

[TABLE]

the result follows. ∎

Proposition 2.9 is used to obtain bounds on the radius of a UCG.

Corollary 2.10**.**

If $H$ is a uniform central graph with $\langle\mathcal{CP}(H)\rangle=P$, $\kappa=\text{cov}_{A}(P)$ and $|\mathcal{I}(H)|\leq s\kappa+t$ where $s,t\in\mathbb{N}$ and $t<\kappa$, then $r(H)\leq s+1$.

Proof.

Observing $(s+1)\kappa>s\kappa+t$ the result follows from the contrapositive of proposition 2.9. ∎

We now construct a graph with center $C$ and centered periphery $P$. Let $C$ and $P$ be graphs, and $\rho\in\mathbb{N}$. Suppose $\overline{P}=\{P_{1},\ldots,P_{k}\}$ is a covering of $P$. Define a graph $G=\mathcal{G}(C,P,\overline{P},\rho)$ as follows:

[TABLE]

and $ab$ is an edge in $G$ if and only if one of the following occurs

$ab$ is an edge of $C$. 2. 2.

$ab$ is an edge of $P$. 3. 3.

$a$ is a vertex of $C$ and $b=x_{i,1}$ for some $i$ with $0\leq i\leq k$. 4. 4.

$a\in P_{i}$ and $b=x_{i,\rho}$ for some $i$ with $1\leq i\leq k$. 5. 5.

$a=x_{i,j}$ and $b=x_{i,j+1}$ for $i$ and $j$ with $0\leq i\leq k,1\leq j\leq\rho-1$

Proposition 2.11**.**

For two given graphs $C$ and $P$, let $\overline{P}=\{P_{1},\ldots,P_{k}\}$ be a covering of $P$ satisfying condition A, and $\rho\geq\text{min}\{\text{diam}(C),2\}$. Then $G=\mathcal{G}(C,P,\overline{P},\rho)$ is a UCG with radius $\rho+1$, $\langle\mathcal{Z}(G)\rangle=C$ and $\langle\mathcal{CP}(G)\rangle=P$.

Proof.

From the construction of $G$, we can show the following:

1. i)

For all $j=1,\dots,\rho$ and any vertex $p$ in $P$, $d(x_{0,j},p)\geq\rho+2$, and so $e(x_{0,j})\geq\rho+2$ and $e(p)\geq\rho+2$. 2. ii)

For all $i$ and $j$ with $1\leq i\leq k$ and $2\leq j\leq\rho$, $d(x_{i,j},x_{0,\rho})\geq\rho+2$, and so $e(x_{i,j})\geq\rho+2$. 3. iii)

Since the covering $\overline{P}$ satisfies condition A, for each $i$ with $1\leq i\leq k$ there exists a vertex $p$ in $P_{j}$ for some $j\neq i$ satisfying $d_{P}(P_{i},p)\geq 2$. Thus $d(x_{i,1},p)\geq\rho+2$ and $e(x_{i,1})\geq\rho+2$. 4. iv)

For a vertex $c$ in $C$, $d(c,x_{i,j})\leq\rho$, $d(c,p)=\rho+1$ for all $p$ in $P$, and $d(c,c^{\prime})\leq 2$ for any vertex $c^{\prime}$ in $C$. This implies $e(c)=\rho+1$ and $EC(c)=V(P)$.

Therefore, $G=\mathcal{G}(C,P,\overline{P},\rho)$ is a UCG with radius $\rho+1$, $\langle\mathcal{Z}(G)\rangle=C$ and $\langle\mathcal{CP}(G)\rangle=P$.

∎

We end this section with a result about conditions when a spanning subgraph of a UCG is still a UCG.

Lemma 2.12**.**

Let $H$ be a UCG with $V(C)=\mathcal{Z}(H)$ and let $G$ be a spanning subgraph of $H$. Then $G$ is a UCG with $V(C)=\mathcal{Z}(G)$ and $\mathcal{CP}(G)=\mathcal{CP}(H)$ if for all $c\in V(C)$ and $x\in V(G)$, $d_{G}(c,x)=d_{H}(c,x)$.

Proof.

Since $E(G)\subset E(H)$, the eccentricity of a vertex $v$ in $G$ are greater than or equal to the eccentricity of $v$ in $H$. The assumption that $d_{G}(c,x)=d_{H}(c,x)$ for all $c\in V(C)$ and $x\in V(G)$ implies $\mathcal{Z}(G)=V(C)$ and $EC(c)$ in $G$ is $\mathcal{CP}(H)$. Hence, $G$ is a UCG with $V(C)=\mathcal{Z}(G)$ and $\mathcal{CP}(G)=\mathcal{CP}(H)$. ∎

3 When $C=K_{n}$ with $n\geq 2$

In this section we compute appendage numbers when the center $C$ is a complete graph $K_{n}$ with $n\geq 2$ in terms of the size of a smallest covering for the centered periphery.

We introduce a new condition on coverings. Let $P$ be a graph with covering $\overline{P}=\{P_{1},\ldots,P_{k}\}$. We say $(P,\overline{P})$ satisfies

Condition B:

if for every $1\leq i\leq k$ and for each $p\in P_{i}$ either

1. 1.

there is a vertex $p^{\prime}\not\in P_{i}$ with $d(p,p^{\prime})\geq 3$, or 2. 2.

there is a $j\neq i$ satisfying $d(p,P_{j})\geq 2$.

Once again, we say the covering $\overline{P}$ satisfies condition B when there is no confusion about $P$. Also define $\text{cov}_{AB}(P)$ to be the smallest size of a covering of $P$ satisfying both condition A and condition B.

Before determining appendage numbers it is necessary to prove the existence of coverings satisfying conditions A and B.

Lemma 3.1**.**

Let $P$ be a graph with $r(P)\geq 2$. Then there exists a covering of $P$ satisfying conditions A and B.

Proof.

Let $V(P)=\{p_{1},\ldots,p_{k}\}$. Then $\overline{P}=\{\{p_{1}\},\ldots,\{p_{k}\}\}$ is a covering of $P$ and $\overline{P}$ satisfies conditions $A$ and $B$ because $r(P)\geq 2$. ∎

We now find the appendage numbers in terms of $\kappa=\text{cov}_{A}(P)$.

Proposition 3.2**.**

$\kappa\leq A_{ucg}(K_{n},P)\leq\kappa+1$* where $\kappa=\text{cov}_{A}(P)$.*

Proof.

By lemma 3.1 there is a covering of $P$ satisfying condition A, so let $\overline{P}=\{P_{1},P_{2},\ldots,P_{\kappa}\}$ be a smallest covering of $P$ satisfying condition A. By proposition 2.11, the graph $G=\mathcal{G}(K_{n},P,\overline{P},1)$ is a UCG with radius 2 and $|\mathcal{I}(G)|=\kappa+1$, and thus $A_{ucg}(K_{n},P)\leq\kappa+1$. The lower bound is from proposition 2.9.

∎

Theorem 3.3**.**

Let $\kappa=\text{cov}_{A}(P)$. Then $A_{ucg}(K_{n},P)=\kappa$ if and only if $\text{cov}_{AB}(P)=\kappa$.

Proof.

Suppose $\text{cov}_{AB}(P)=\kappa$ and let $\overline{P}=\{P_{1},\ldots,P_{\kappa}\}$ be a smallestl covering of $P$ with respect to conditions A and B. Also, let $G=\mathcal{G}(K_{n},P,\overline{P},1)-\{x_{0,1}\}$. Since $\overline{P}$ satisfies condition B, one can verify that for $G$, $e(p)\geq 3$ for all $p\in V(P)$. Using a similar argument to proposition 2.11 we can show $G$ is a UCG with $r(G)=2$, $\langle\mathcal{CP}(G)\rangle=P$, $\langle\mathcal{Z}(G)\rangle=K_{n}$ and $|\mathcal{I}(G)|=\kappa$.

Now assume $A_{ucg}(K_{n},P)=\kappa$. Let $H$ be a UCG with $P=\langle\mathcal{CP}(H)\rangle,\langle\mathcal{Z}(H)\rangle=K_{n}$ and $|\mathcal{I}(H)|=\kappa$. From corollary 2.10, $r(H)=2$. Let $\{P_{1},\ldots,P_{k}\}$ be the induced covering of $P$ as in lemma 2.2. We know

[TABLE]

and so $k=\kappa$. Let $D_{1}=\{x_{1},\ldots,x_{k}\}$, where $x_{i}$ is associated to $P_{i}$.

Suppose $p$ is a vertex in $P_{i}$. Since $e(p)\geq 3$ there is an $x\in V(H)$ with $d_{H}(p,x)\geq 3$. Note, $x\not\in P_{i}$, because all vertices in $P_{i}$ are adjacent to $x_{i}$. If $x\in P_{j}$ for some $j\neq i$, then condition B-1 is satisfied by definition. Finally, if $x=x_{j}\in D_{1}$ for some $j\neq i$, then $d_{P}(p,P_{j})\geq 2$ and so $\overline{P}$ satisfies condition B-2. ∎

In section 5 we determine when $\text{cov}_{A}(P)=\text{cov}_{AB}(P)$.

4 When $C\neq K_{n}$.

For this section assume that $C$ is not a complete graph and so $\text{diam}(C)\geq 2$. The results in this section mirror those of when $C=K_{n}$, however conditions on the coverings of $P$, as well as the proofs, are more technical.

Proposition 4.1**.**

For a given pair of graphs $(C,P)$ , $2\kappa\leq A_{ucg}(C,P)\leq 2\kappa+2$ where $\kappa=\text{cov}_{A}(P)$.

Proof.

For a given pair of graphs $C$ and $P$, suppose $H$ is a UCG with $\langle\mathcal{Z}(H)\rangle=C$ and $\langle\mathcal{CP}(H)\rangle=P$. Then $r(H)\geq\text{diam}(C)+1\geq 3$. From corollary 2.9 we obtain a lower bound $A_{ucg}(C,P)\geq 2\kappa$.

For the upper bound consider $G=\mathcal{G}(C,P,\overline{P},2)$ where $\overline{P}$ is a smallest covering of $P$ satisfying condition A. By proposition 2.11 $G$ is a UCG with $|\mathcal{I}(G)|=2\kappa+2$ and so $A_{ucg}(C,P)\leq 2\kappa+2$. ∎

In the proof of proposition 4.1 we use the fact that $\mathcal{G}(C,P,\overline{P},2)$ is a UCG so long as $\overline{P}$ satisfies condition A. We now consider a modification of this graph, and determine conditions that this new graph is a UCG.

Let $\overline{P}=\{P_{1},\ldots,P_{k}\}$ be a covering of $P$ and let

[TABLE]

By construction, $d(c,p)=3$ for all $c\in V(C)$, $p\in V(P)$, and $d(c,x)\leq 2$ for $x\in V(G)-V(P)$. So $G$ is a UCG with center $C$ and centred periphery $P$ if and only if for all $u\in V(G)-V(C)$, $e(u)\geq 4$.

Consider the case $u=x_{i,1}$. If $G$ is a UCG, $\overline{P}$ must satisfy condition A. This implies $e(x_{i,1})\geq 4$ for all $1\leq i\leq k$.

Next, consider the case $u=x_{i,2}$ for some $1\leq i\leq k$. Since $e(u)\geq 4$ there exists a $v\in V(G)$ such that $d(u,v)\geq 4$. Because $v\not\in V(C)$ and $d(u,x_{j,1})\leq 3$, either $v\in V(P)$ or $v=x_{j,2}$ for some $j\neq i$. If $p\in V(P)$ then $v\not\in P_{i}$ and $d(v,P_{i})\geq 3$. If $v=x_{j,2}$, then $d(P_{i},P_{j})\geq 2$.

Finally, consider the case $u\in P_{i}$ for some $i$. If $G$ is a UCG, then $e(u)\geq 4$, and so there is a $v\in V(G)$ such that $d(u,v)\geq 4$. Since $v\not\in V(C)$ and $v\not\in P_{i}$ because $d(p,p^{\prime})\leq 2$ for all $p,p^{\prime}\in P_{i}$, it follows that either $v=x_{j,1}$, $v=x_{j,2}$ or $v\in P_{j}$ for some $j\neq i$. If $v=x_{j,1}$ then $d(u,P_{j})\geq 2$ for $j\neq i$. If $v=x_{j,2}$ then $d(u,P_{j})\geq 3$. Finally, if $v\in P_{j}$ for some $j\neq i$, then for all $p\in P_{j}$

[TABLE]

We conclude $d(u,P_{j})\geq 2$. This analysis gives rise to two new conditions on $\overline{P}$.

Let $\overline{P}=\{P_{1},\ldots,P_{k}\}$ be a covering of a graph $P$. We say $(P,\overline{P})$ satisfies if for all $1\leq i\leq k$

Condition A′:**

if either

1. 1.

there is a $p\not\in P_{i}$ satisfying $d(P_{i},p)\geq 3$, or 2. 2.

there is a $j\neq i$ satisfying $d(P_{i},P_{j})\geq 2$.

Condition B′:**

if for each $p\in P_{i}$ there is a $j\neq i$ satisfying $d(p,P_{j})\geq 2$.

Condition A*′* arises from $e(x_{i,2})\geq 4$ and condition B*′* from $e(p)\geq 4$, for $p\in V(P)$. Note condition A*′* implies condition A, and condition B*′* implies condition B. We often abuse notation and say the covering $\overline{P}$ satisfies a specified condition.

Let $\text{cov}_{A^{\prime}}(P)$ be the smallest size of the covering of $P$ satisfying condition A*′* and $\text{cov}_{A^{\prime}B^{\prime}}(P)$ the smallest size of the covering of $P$ satisfying conditions A*′* and B*′*.

The arguments used to determine conditions A*′* and B*′* are bi-directional. This implies the following proposition.

Proposition 4.2**.**

Let $C$ and $P$ be graphs, $\overline{P}$ a covering of $P$, and $G=\mathcal{G}(C,P,\overline{P},2)-\{x_{0,1},x_{0,2}\}$. Then $G$ is a UCG with center $C$ and centered periphery $P$ if and only if $\overline{P}$ satisfies conditions A′ and B′.

We are now ready to relate appendage numbers to coverings.

Proposition 4.3**.**

Let $\kappa=\text{cov}_{A}(P)$. For a graph $P$ with $r(P)>1$, $\text{cov}_{A^{\prime}B^{\prime}}(P)=\kappa$ if and only if $A_{ucg}(C,P)=2\kappa$.

Proof.

First, assume $\text{cov}_{A^{\prime}B^{\prime}}(P)=\kappa$. Let $\overline{P}=\{P_{1},\ldots,P_{\kappa}\}$ be a covering of $P$ satisfying conditions A*′* and B*′*. Then by proposition 4.2 the graph

[TABLE]

is a UCG and $|\mathcal{I}(G)|=2\kappa$ and so $A_{ucg}(C,P)\leq 2\kappa$. However, by proposition 4.1, $A_{ucg}(C,P)\geq 2\kappa$ and the result follows.

Next, assume $A_{ucg}(C,P)=2\kappa$. Let $H$ be a UCG with $P=\langle\mathcal{CP}(H)\rangle$, $C=\langle\mathcal{Z}(H)\rangle$ and $|\mathcal{I}(H)|=2\kappa$. Note that $r(H)=3$ by corollary 2.10.

Let $\{P_{1},\ldots,P_{k}\}$ be the induced covering of $P$ as in lemma 2.2. There exists a subcover $\overline{P}=\{P_{1},\ldots,P_{k^{\prime\prime}}\}$ and an associated set of vertices $\{\tilde{p}_{1},\ldots,\tilde{p}_{k^{\prime\prime}}\}$ as in lemma 2.1. Assume $D_{1}^{\prime\prime}=\{x_{1},\ldots,x_{k^{\prime\prime}}\}$, and by corollary 2.6 $|D_{2}^{\prime\prime}|\geq k^{\prime\prime}$. Because $\kappa=\text{cov}_{A}(P)$ and $\overline{P}$ satisfies condition A we know

[TABLE]

and so $k^{\prime\prime}=|D_{1}^{\prime\prime}|=|D_{2}^{\prime\prime}|=\kappa$. This implies $D_{1}=D_{1}^{\prime\prime},D_{2}=D_{2}^{\prime\prime}$ and $\kappa=k^{\prime\prime}=k$.

We now show $H$ contains a spanning subgraph isomorphic to

[TABLE]

First, for each $\tilde{p}_{i}$ defined above and each central vertex $c$ there is a $(c,\tilde{p}_{i})$-radial path. By construction of $\tilde{p}_{i}$ this path must contain $x_{i}$, and thus $x_{i}$ is adjacent to every $c$ in the center by lemma 2.8.

By lemma 2.5 there is an enumeration of $D_{2}=\{y_{1},\ldots,y_{k}\}$ so that $y_{i}$ is adjacent to both $x_{i}$ and each vertex in $P_{i}$. Therefore, $G=\mathcal{G}(C,P,\overline{P},2)-\{x_{0,1},x_{0,2}\}$ is isomorphic to a spanning subgraph of $H$. By lemma 2.12, $G$ is a UCG and by proposition 4.2, $\overline{P}$ satisfies conditions A*′* and B*′*.

∎

We now move on to understand when $A_{ucg}(C,P)=2\kappa+1$. Let $H$ be a UCG such that $\langle\mathcal{Z}(H)\rangle=C$, $\langle\mathcal{CP}(H)\rangle=P$ and $|\mathcal{I}(H)|=2\kappa+1$. By proposition 4.3, $\kappa\neq\text{cov}_{A^{\prime}B^{\prime}}(P)$. Also, by corollary 2.10 $r(H)=3$, and hence $|D_{1}|+|D_{2}|=2\kappa+1$. Let $\{P_{1},\ldots,P_{k}\}$ be the induced covering of $P$ through $D_{1}=\{x_{1},\ldots,x_{k}\}$ as in lemma 2.2. Without loss of generality let $\overline{P}=\{P_{1},\ldots,P_{k^{\prime\prime}}\}$, with $k^{\prime\prime}\leq k$, be a subcover with an associated set of vertices $\{\tilde{p}_{1},\ldots,\tilde{p}_{k^{\prime\prime}}\}$ as in lemma 2.1. Then

[TABLE]

and hence $\kappa=k^{\prime\prime}$.

Since

[TABLE]

either $|D_{1}|=\kappa+1$ or $|D_{2}|=\kappa+1$.

We first address when $|D_{1}|=\kappa+1$ and $|D_{2}|=\kappa$.

Proposition 4.4**.**

Assume $A_{ucg}(C,P)=2\kappa+1$, and let $H$ be a UCG such that $\langle\mathcal{Z}(H)\rangle=C$, $\langle\mathcal{CP}(H)\rangle=P$, and $|\mathcal{I}(H)|=2\kappa+1$ where $\kappa=\text{cov}_{A}(P)$. Furthermore, assume $|D_{1}|=\kappa+1$. Then $\text{cov}_{A^{\prime}}(P)=\kappa$.

Proof.

By corollary 2.6 it follows that

[TABLE]

Therefore $D_{2}=D_{2}^{\prime\prime}$ and $|D_{1}^{\prime\prime}|=\kappa$. Since $k=|D_{1}|=\kappa+1$, $x_{k}\in D_{1}$ but $x_{k}\not\in D_{1}^{\prime\prime}$.

Next, we prove $x_{k}\not\in D_{1}^{\prime}$ by showing if $x_{k}\in D_{1}^{\prime}$, then $\overline{P}$ satisfies both conditions A*′* and B*′*. Hence $A_{ucg}(C,P)=2\kappa$ by proposition 4.3.

Assume $x_{k}\in D_{1}^{\prime}$, that is $D_{1}=D_{1}^{\prime}$. Because $|D_{2}|=|D_{2}^{\prime\prime}|=\kappa=|D_{1}^{\prime\prime}|$ we may assume $D_{2}=\{y_{1},\ldots,y_{\kappa}\}$ such that each $y_{i}$ is adjacent to $x_{i}$ and all vertices of $P_{i}$ by lemma 2.7. To understand the structure of $P_{k}$ define the indexing set

[TABLE]

From the definition of $D_{1}^{\prime}$ and $I$,

[TABLE]

For each $j\not\in I$ every radial path to $\tilde{p}_{j}$ must contain $x_{j}$. Hence, each central vertex $c\in V(C)$ is adjacent to $x_{j}\in D_{1}$ for $j\not\in I$. A similar argument shows that for each $c\in V(C)$ and each $i\in I$, $c$ is adjacent to either $x_{k}$ or $x_{i}$.

We now show $\overline{P}$ satisfies conditions A*′* and B*′*.

First, assume condition B*′* fails. Then there exists an $\iota$ and a vertex $p\in P_{\iota}$ such that $d_{P}(p,P_{j})\leq 1$ for all $j\neq\iota$. Therefore, there is a $p_{j}\in P_{j}$ such that $d(p,p_{j})\leq 1$, and so the following hold:

1. i)

$d(p,x_{j})\leq d(p,p_{j})+d(p_{j},x_{j})\leq 3$. 2. ii)

$d(p,y_{j})\leq d(p,p_{j})+d(p_{j},y_{j})\leq 2$. 3. iii)

$d(p,p^{\prime})\leq d(p,p_{j})+d(p_{j},p^{\prime})\leq 3$ for all $p^{\prime}\in P_{j}$. 4. iv)

$d(p,x_{\iota})=2$. 5. v)

$d(p,y_{\iota})=1$. 6. vi)

$d(c,p)=3$ for $c\in V(C)$.

If $\iota\in I$, then $d(p,x_{k})=2$. If $\iota\not\in I$, then for a $j\in I$

[TABLE]

since $d(p_{j},x_{k})=2$. Therefore $e(p)=3$, a contradiction, and so $\overline{P}$ satisifies condition B*′*.

Next, assume condition A*′* fails. Then there exists an $\iota$ such that $d_{P}(P_{\iota},p^{\prime})\leq 2$ for all $p^{\prime}\in V(P)-P_{i}$ and $d_{P}(P_{\iota},P_{j})\leq 1$ for all $j\neq\iota$. Then $d(y_{\iota},p)\leq 3$ for all $p\in V(P)$ and $d(y_{\iota},y_{j})\leq 3$ for all $j\neq\iota$. We obtain a contradiction by showing $y_{\iota}$ is in the center.

If $\iota\not\in I$ then for each $x_{j}\in D_{1}$ there is a $c\in V(C)$ that is adjacent to $x_{j}$. Then $y_{\iota}-x_{\iota}-c-x_{j}$ is a path and $d(y_{\iota},x_{j})\leq 3$. Similarly when $\iota\in I$ and $j\not\in I$, $d(y_{\iota},x_{j})\leq 3$. If $\iota,j\in I$ then $y_{\iota}-x_{k}-y_{j}-x_{j}$ is a path and so $d(y_{\iota},x_{j})\leq 3$. Finally, $d(y_{\iota},c)=2$ for all $c\in V(C)$, and so $e(y_{\iota})=3$, a contradiction and $\overline{P}$ satisfies condition A*′*.

Since $\overline{P}$ satisfies conditions A*′* and B*′*, $A_{ucg}(C,P)=2\kappa$ by proposition 4.3. This contradicts the assumptions of the proposition and hence $x_{k}\not\in D_{1}^{\prime}$ and so $D_{1}^{\prime}=D_{1}^{\prime\prime}$.

Then $x_{k}$ is not on a radial path because $x_{k}\not\in D_{1}^{\prime}$. Therefore, vertices adjacent to $x_{k}$ are in $C$ or $D_{1}^{\prime\prime}$. Furthermore each $y_{i}\in D_{2}$ satisfies $e(y_{i})\geq 4$ from $H$ being a UCG with $r(H)=3$ . Therefore, there exists a $u\in V(G)$ with $d(y_{i},u)\geq 4$. Because $d(c,y_{i})=2$ for all $c\in V(C)$, we know $u\not\in V(C)$.

Because $D_{1}^{\prime}=D_{1}^{\prime\prime}$, each $(c,\tilde{p}_{i})$-radial path contains $x_{i}$, and hence each $c$ is adjacent to $x_{i}$. It follows that for a $c\in V(C)$ adjacent to $x_{j}\in D_{1}$ that

[TABLE]

which means $u\not\in D_{1}$.

If $u\in V(P)$ then $d(u,P_{i})\geq 3$ and condition A*′-1 is satisfied for $i$. If $u=y_{j}\in D_{2}^{\prime\prime}$ then $d_{P}(P_{i},P_{j})\geq 2$ and condition A′-2 is satisfied for $i$. Since these hold for each $i$, $\overline{P}$ satisfies condition A′* and $\text{cov}_{A^{\prime}}(P)=\kappa$. ∎

A weak converse of proposition 4.4 also holds.

Proposition 4.5**.**

If $P$ is a graph with $r(P)>1$, $\kappa=\text{cov}_{A}(P)\neq\text{cov}_{A^{\prime}B^{\prime}}(P)$ and $\text{cov}_{A^{\prime}}(P)=\text{cov}_{A}(P)$, then $A_{ucg}(C,P)=2\kappa+1$.

Proof.

Assume $cov_{A^{\prime}B^{\prime}}(P)\neq\kappa$ but $cov_{A^{\prime}}(P)=\kappa$. Let $\overline{P}=\{P_{1},\ldots P_{\kappa}\}$ be a smallest covering with respect to condition A*′* and

[TABLE]

We claim $G$ is a UCG with $C=\langle\mathcal{Z}(G)\rangle$, $P=\langle\mathcal{CP}(G)\rangle$ and $|\mathcal{I}(G)|=2\kappa+1$.

By proposition 4.2 the graph $G^{\prime}=G-\{x_{0,1}\}$ is a UCG if and only if $\overline{P}$ satisfies conditions A*′* and B*′. In $G^{\prime}$, if $\overline{P}$ satisfies conditions A′*, then $e(x_{i,1})\geq 4$ and $e(x_{i,2})\geq 4$. This is still holds in $G$. Furthermore, for each vertex $p\in V(P)$, $d(x_{0,1},p)=4$ in $G$, and so $G$ is a UCG. Therefore, $A_{ucg}(C,P)\leq 2\kappa+1$, but by proposition 4.3 $A_{ucg}(C,P)>2\kappa$. ∎

The case when $|D_{2}|=\kappa+1$ is more complicated. To understand this case we consider a new graph $G$, and determine new conditions for when $G$ is a UCG.

For two graphs $C$ and $P$, let $\overline{P}=\{P_{1},\ldots,P_{k}\}$ be a covering of $P$, and $\overline{Q}=\{Q_{0},Q_{1},P_{2},\ldots,P_{k}\}$ be a covering such that $Q_{0}\cup Q_{1}=P_{1}$.

Define a graph $G=\mathcal{G}^{\prime}(C,P,\overline{Q})$ as follows (see figure 3):

[TABLE]

and $ab$ is an edge in $G$ if and only if one of the following occurs

1. i)

$ab$ is an edge of $C$. 2. ii)

$ab$ is an edge of $P$. 3. iii)

$a$ is a vertex of $C$ and $b=x_{i}$ for some $i$ with $1\leq i\leq k$. 4. iv)

$a\in P_{j}$ and $b=y_{j}$ for some $j$ with $2\leq j\leq k$. 5. v)

$a\in Q_{l}$ and $b=y_{l}$ for some $l=0,1$. 6. vi)

$a=x_{i}$ and $b=y_{i}$ for $1\leq i\leq k$ 7. vii)

$a=x_{1}$ and $b=y_{0}$.

We now determine conditions on $\overline{Q}$ for $G$ to be a UCG with $C=\langle\mathcal{Z}(G)\rangle$ and $P=\langle\mathcal{CP}(G)\rangle$. By construction for all $c\in V(C)$ and $p\in V(P)$, $d(c,p)=3$, and $d(c,x)\leq 2$ for all $x\in V(G)-V(P)$. So $G$ is a UCG with center $C$ and centred periphery $P$ if and only if for all $u\in V(G)-V(C)$, $e(u)\geq 4$. Then there exists a vertex $v$ with $d(u,v)\geq 4$.

First consider the case $u=x_{i}$. In every UCG the induced covering satisfies condition A. This implies $v\in V(P)$ and $e(x_{i})\geq 4$ for all $1\leq i\leq k$. Note that, in $G$ the induced covering is $\overline{P}$, not $\overline{Q}$.

Next, consider the case $u=y_{i}$ for some $2\leq i\leq k$. Since $v\not\in V(C)$ and $d(v,x_{j})\leq 3$ for $1\leq j\leq k$, either $v\in V(P)$ or $v=y_{j}$ for $0\leq j\leq k$ and $j\neq i$. The one of the following holds.

1. i)

If $v\in V(P)$ then $v\not\in P_{i}$ and $d(P_{i},v)\geq 3$. 2. ii)

If $v=y_{j}$, $2\leq j\leq k$ and $j\neq i$, then $d(P_{i},P_{j})\geq 2$. 3. iii)

If $v=y_{l}$ for $l=0$ or $1$, then $d(P_{i},Q_{l})\geq 2$.

Now, consider the case $u=y_{0}$ or $y_{1}$. Without loss of generality we assume $u=y_{0}$. Since $v\not\in V(C)$, $v\neq x_{j}$ for $1\leq j\leq k$, $v\neq y_{1}$ and $v\not\in Q_{0}\cup Q_{1}=P_{1}$, the following must hold.

1. i)

If $v\in V(P)$ then $v\not\in P_{1}$ and $d(Q_{0},v)\geq 3$. 2. ii)

If $v=y_{j}$, $2\leq j\leq k$ and $j\neq i$, then $d(Q_{0},P_{j})\geq 2$.

Now, consider the case $u=p\in P_{i}$ for some $2\leq i\leq k$. Once again, if $G$ is a UCG, then $e(p)\geq 4$, and so there exists a $v\in V(G)$ such that $d(u,v)\geq 4$. We know $v\not\in V(C)$. Also, $v\not\in P_{i}$ since $d(p,p^{\prime})\leq 2$ for all $p^{\prime}\in P_{i}$. Also note, that for $j\geq 2$ and $j\neq i$, if $d(u,y_{j})\geq 4$ then $d(u,x_{j})\geq 4$. This implies that we do not need to determine the conditions for $d(u,y_{j})\geq 4$. Given this, one of the following must hold.

1. i)

If $v=x_{j}$ for $j\neq i$ and $2\leq j\leq k$, then $d(u,P_{j})\geq 2$. 2. ii)

If $v=x_{1}$, then $d(u,Q_{0})\geq 2$ and $d(u,Q_{1})\geq 2$. 3. iii)

If $v=y_{l}$ for $l=0$ or $1$, then $d(u,Q_{l})\geq 3$. 4. iv)

If $v=p^{\prime}\in P_{j}-P_{i}$ for $2\leq j\leq k$ and $j\neq i$, then $d(u,p^{\prime})\geq 4$ and $d(u,P_{j})\geq 2$. 5. v)

If $v=p^{\prime}\in Q_{l}-P_{i}$ for $l=0$ or $1$, then $d(p,Q_{l})\geq 2$ and $d(u,p^{\prime})\geq 4$.

The last case to consider, without loss of generality, is $u=p\in Q_{0}$. Since $d(u,v)\leq 3$ if $v\in V(C)$, $v=x_{1},y_{0}$ or $y_{1}$ or $v\in Q_{0}$, the following must hold.

1. i)

If $v=x_{j}$, then $2\leq j\leq k$ and $d(u,P_{j})\geq 2$. 2. ii)

If $v=y_{j}$ for $2\leq j\leq k$, then $d(u,P_{j})\geq 3$. 3. iii)

If $v=p^{\prime}\in P_{j}-Q_{0}$ for $2\leq j\leq k$, then $d(u,P_{j})\geq 2$ and $d(u,p^{\prime})\geq 4$. 4. iv)

If $v=p^{\prime}\in Q_{1}-Q_{0}$, then $d(u,p^{\prime})\geq 4$ and $d(u,Q_{1})\geq 2$.

The above discussion is in terms of $\overline{Q}$, however the rest of the paper is in terms of $\overline{P}$. For this reason now summarize the discussion in terms of two technical conditions on $\overline{P}$.

Let $\overline{P}=\{P_{1},\ldots,P_{k}\}$ be a covering of a graph $P$. For a given $\iota$ and two sets $Q_{0}$ and $Q_{1}$ such that $Q_{0}\cup Q_{1}=P_{\iota}$, let

[TABLE]

We say $(P,\overline{P},\overline{Q})$ satisfies

Condition A′′:**

1. 1.

if for each $i\neq\iota$ one of the following holds

1. (a)

there is a $p\not\in P_{i}$ satisfying $d(P_{i},p)\geq 3$, or 2. (b)

there is a $j\neq\iota$ satisfying $d(P_{i},P_{j})\geq 2$, or 3. (c)

there is an $l=0$ or $1$ such that $d(P_{i},Q_{l})\geq 2$, 2. 2.

and if for all $l=0,1$ either

1. (a)

there is a $p\not\in P_{\iota}$ satisfying $d(Q_{l},p)\geq 3$, or 2. (b)

there is a $j\neq\iota$ satisfying $d(Q_{l},P_{j})\geq 2$.

Condition B′′:**

1. 1.

if for each $p\in P_{i}$, $i\neq\iota$ one of the following holds

1. (a)

there is a $j\neq\iota$ satisfying $d(p,P_{j})\geq 2$, or 2. (b)

$d(p,Q_{0})\geq 2$ and $d(p,Q_{1})\geq 2$, or 3. (c)

there is an $l=0$ or $1$ such that $d(p,Q_{l})\geq 3$, or 4. (d)

there $l=0$ or $1$ such that $d(p,Q_{l})\geq 2$ and a $p^{\prime}\in Q_{l}$ so that $d(p,p^{\prime})\geq 4$. 2. 2.

and if for each $l=0,1$ and each $p\in Q_{l}$, either

1. (a)

there is a $j\neq\iota$ satisfying $d(p,P_{j})\geq 2$, or 2. (b)

there exists $p^{\prime}\in P_{\iota}-Q_{l}$ such that $d(p,p^{\prime})\geq 4$ and $d(p,Q_{l^{\prime}})\geq 2$, where $l^{\prime}=0$ or $1$ but $l^{\prime}\neq l$.

When there exists an $\iota$, $Q_{0}$ and $Q_{1}$ such that $(P,\overline{P},\overline{Q})$ satisfies condition A*′′* (resp. condition B*′′), we say $(P,\overline{P})$ or simply $\overline{P}$ satisfies condition A′′* (condition B*′′). Without loss of generality we may renumber $\overline{P}$ so that $\iota=1$. Note that A′* and B*′* imply A*′′* and B*′′* by taking $\iota=1$ and letting $Q_{0}=P_{1}$ and $Q_{1}=\emptyset$. However, condition A*′′* does not imply condition A. So let $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)$ be the smallest size of the covering $\overline{P}$ of $P$ satisfying conditions A, A*′′* and B*′′*.

Similar to the discussion of conditions A*′* and B*′, the arguments used to determine conditions A′′* and B*′′* from the graph $\mathcal{G}(C,P,\overline{Q})$ are bi-directional. We summarize the discussion in the following proposition.

Proposition 4.6**.**

For graphs $C$ and $P$, and a triple $(P,\overline{P},\overline{Q})$, $G=\mathcal{G}^{\prime}(C,P,\overline{Q})$ is a UCG if and only if $\overline{P}$ satisfies condition A and $(P,\overline{P},\overline{Q})$ satisfies conditions A′′ and B′′.

We are now ready to prove analogues to propositions 4.4 and 4.5.

Proposition 4.7**.**

Assume $A_{ucg}(C,P)=2\kappa+1$, and let $H$ be a UCG with $\langle\mathcal{Z}(H)\rangle=C$, $\langle\mathcal{CP}(H)\rangle=P$, and $|\mathcal{I}(H)|=2\kappa+1$ where $\kappa=\text{cov}_{A}(P)$. Furthermore, assume $|D_{2}|=\kappa+1$. Then $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=\kappa$.

Proof.

Let $H$ be a UCG with $C=\langle\mathcal{Z}(H)\rangle$, $P=\langle\mathcal{CP}(H)\rangle$, $|\mathcal{I}(H)|=2\kappa+1$ and $|D_{2}|=\kappa+1$. Since $|D_{1}|=\kappa$ and $\kappa\leq|D_{1}^{\prime\prime}|\leq|D_{1}|$, it follows that $|D_{1}|=|D_{1}^{\prime\prime}|$. We prove this proposition by studying the structure of a spanning subgraph.

Let $D_{1}=\{x_{1},\ldots,x_{\kappa}\}$ and $\{\tilde{p}_{1},\ldots,\tilde{p}_{\kappa}\}$ be a set of vertices associated to the induced cover $\overline{P}$. By proposition 2.5 there exists an enumeration $\{y_{0},\ldots,y_{\kappa}\}$ of $D_{2}$ such that the vertex $y_{j}$ is adjacent to $x_{j}$ and $\tilde{p}_{j}$ for each $j$, $1\leq j\leq\kappa$. We may also assume $y_{0}$ is adjacent to $x_{1}$.

We now define a different cover of $P$. For each $i$, $0\leq i\leq\kappa$, let

[TABLE]

Let $P_{1}^{\prime}=Q_{0}\cup Q_{1}$ and $P_{i}^{\prime}=Q_{i}$ for $2\leq i\leq\kappa$, $\overline{P}^{\prime}=\{P_{1}^{\prime},\ldots,P_{\kappa}^{\prime}\}$ and $\overline{Q}=\{Q_{0},\ldots,Q_{\kappa}\}$. Since $|D_{1}|=|D_{1}^{\prime\prime}|$ every vertex of $C$ is adjacent to vertex in $D_{1}$ by lemma 2.8. Therefore, $G=\mathcal{G}^{\prime}(C,P,\overline{Q})$ is isomorphic to a spanning subgraph of $H$, and is a UCG by lemma 2.12. By proposition 4.6 $\overline{P}^{\prime}$ satisfies conditions A, A*′′* and B*′′* which means

[TABLE]

We conclude $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=\kappa$.

∎

Proposition 4.8**.**

If $P$ is a graph with $r(P)>1$, $\kappa=\text{cov}_{A}(P)\neq\text{cov}_{A^{\prime}B^{\prime}}(P)$ and $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=\text{cov}_{A}(P)$, then $A_{ucg}(C,P)=2\kappa+1$.

Proof.

Assume $cov_{A^{\prime}B^{\prime}}(P)\neq\kappa$ but $cov_{AA^{\prime\prime}B^{\prime\prime}}(P)=\kappa$. By proposition 4.3 $A_{ucg}(C,P)>2\kappa$, so we need to show $A_{ucg}(C,P)\leq 2\kappa+1$. Let $\overline{P}$ be a smallest covering of $P$ with respect to condition A such that there is a refined cover $\overline{Q}$ where the pair $(\overline{P},\overline{Q})$ is smallest with respect to conditions A*′′* and B*′′*. Without loss of generality we may assume $\iota=1$. By proposition 4.6 $G=\mathcal{G}(C,P,\overline{Q})$ is a UCG with $C=\langle\mathcal{Z}(G)\rangle$, $P=\langle\mathcal{CP}(G)\rangle$ and $|\mathcal{I}(G)|=2\kappa+1$, which implies $A_{ucg}(C,P)\leq 2\kappa+1$. ∎

The following theorem summarizes propositions 4.1, 4.3, 4.4, 4.5, 4.7, and 4.8.

Theorem 4.9**.**

If $P$ is a graph with $r(P)>1$ and $\text{cov}_{A}(P)=\kappa$, then following holds:

$A_{ucg}(C,P)=2\kappa$* if and only if $\text{cov}_{A^{\prime}B^{\prime}}(P)=\kappa$.* 2. 2.

$A_{ucg}(C,P)=2\kappa+1$* if and only if $\kappa\neq\text{cov}_{A^{\prime}B^{\prime}}(P)$ and either*

1. (a)

$\text{cov}_{A^{\prime}}(P)=\kappa$* or* 2. (b)

$\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=\kappa$. 3. 3.

$A_{ucg}(C,P)=2\kappa+2$* otherwise.*

5 Coverings

In this section we determine when $\kappa=\text{cov}_{A}(P)$ is the size of smallest coverings with respect to the other conditions described in sections 3 and 4. To do this we introduce one last set of notation. For a graph $G$, a vertex $V$ in $G$ and $s\in\mathbb{N}$ let

[TABLE]

be the closed $s$ neighborhood of $v$. When $s=1$ we simply let $N_{1}[x]=N[x]$.

Proposition 3.2 and theorem 3.3 determine $A_{ucg}(K_{n},P)$ up to knowing when $\text{cov}_{A}(P)=\text{cov}_{AB}(P)$. We now determine conditions for a graph $P$ to satisfy $\text{cov}_{A}(P)=\text{cov}_{AB}(P)$.

Proposition 5.1**.**

If $P$ is a graph with $\text{diam}(P)\geq 3$, then $\text{cov}_{A}(P)=2$.

Proof.

Suppose $x$ is a vertex in $P$ satisfying $e(x)\geq 3$. Let $P_{1}=N[x]$ and $P_{2}=V(P)-P_{1}$. Then $d(x,P_{2})\geq 2$. Since $e(x)\geq 3$, $P_{2}$ contains a vertex $y$ satisfying $d(x,y)\geq 3$, and so $d(y,P_{1})\geq 2$. Thus, $\{P_{1},P_{2}\}$ is a covering of $P$ satisfying condition A. ∎

Proposition 5.2**.**

If $P$ is a graph satisfying $\text{diam}(P)\geq 4$ and $r(P)\geq 3$, then $\text{cov}_{AB}(P)=2$.

Proof.

Let $x$ and $y$ be vertices in $P$ satisfying $d(x,y)=4$. We construct $P_{1}$ and $P_{2}$ recursively. Initialize $P_{1}:=N[x]$ and $P_{2}:=N[y]$. Note that $d(x,N[y])=d(y,N[x])=3$. For a vertex $z$ in $V(P)-(P_{1}\cup P_{2})$, update $P_{1}$ and $P_{2}$ as follows.

1. i)

If there is a vertex $p\in P_{2}$ satisfying $d(z,p)\geq 3$, then let $P_{1}:=P_{1}\cup\{z\}$. 2. ii)

Else if there is a vertex $p\in P_{1}$ satisfying $d(z,p)\geq 3$, then let $P_{2}:=P_{2}\cup\{z\}$. 3. iii)

Else there is a vertex $p\in V(P)-(P_{1}\cup P_{2})$ satisfying $d(z,p)=3$, since $r(P)=3$. Let $P_{1}:=P_{1}\cup\{z\}$ and $P_{2}:=P_{2}\cup\{p\}$.

Continue until all vertices of $P$ have been accounted for. By construction $\{P_{1},P_{2}\}$ is a covering of $P$ satisfying condition B. Since $N[x]\subset P_{1}$, $d(x,P_{2})\geq 2$ and similarly $d(y,P_{1})\geq 2$. Thus $\{P_{1},P_{2}\}$ satisfies condition A. ∎

Proposition 5.3**.**

If $P$ is a graph with $r(P)=2$, then $\text{cov}_{AB}(P)\neq 2$.

Proof.

Suppose there is a covering $\{P_{1},P_{2}\}$ of $P$ satisfying conditions A and B. Assume $P_{1}$ contains a vertex $z$ with $e(z)=2$. By condition B, $d(z,P_{2})=2$ and so $N(z)\cap P_{2}=\emptyset$. If $p$ is a vertex in $P_{2}$, then $d(z,p)=2$ and so $d(P_{1},p)=1$, a contradiction to condition A.

∎

Proposition 5.4**.**

For every $\alpha\in\mathbb{N}$, there exists a graph $P$ with $r(P)=\text{diam}(P)=2$ and $\text{cov}_{A}(P)=2\alpha$.

Proof.

Construct $P^{\alpha}=(V^{\alpha},E^{\alpha})$ as follows. Let

[TABLE]

and

[TABLE]

Then $r(P^{\alpha})=\text{diam}(P^{\alpha})=2$.

We now show $\text{cov}_{A}(P^{\alpha})=2\alpha$. By lemma 3.1 there exists a covering

[TABLE]

of $P^{\alpha}$ satisfying condition A. Suppose $e_{1}\in P^{\alpha}_{1}$. If $P^{\alpha}_{1}$ contains a vertex other than $e_{1}$, then $d(P^{\alpha}_{1},x)=1$ for all $x\in V^{\alpha}-P^{\alpha}_{1}$, a contradiction to condition A. Therefore $P^{\alpha}_{1}=\{e_{1}\}$. Similarly $|P^{\alpha}_{i}|=1$ for all $i=1,2,\ldots,k$ and $\text{cov}_{A}(P^{\alpha})=2\alpha$. ∎

Propositions 5.1 through 5.3 determine when $\text{cov}_{A}(P)=\text{cov}_{AB}(P)$ for all graphs $P$ with $r(P)>1$ except those with $r(P)=\text{diam}(P)=2$ and those with $r(P)=\text{diam}(P)=3$. Proposition 5.4 gives insight into richness of the case $r(P)=\text{diam}(P)=2$. However, we do not have any definitive results for $r(P)=\text{diam}(P)=3$. This is discussed further in section 6.

For a non-complete graph $C$, theorem 4.9 relates $A_{ucg}(C,P)$ to $\text{cov}_{A^{\prime}B^{\prime}}(P)$, $\text{cov}_{A^{\prime}}(P)$, and $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)$. Because of proposition 5.4, we only consider $P$ with $\text{diam}(P)>2$. By proposition 5.1 we need to understand when the smallest coverings are of size $2$.

Proposition 5.5**.**

$P$* is a graph with $\text{cov}_{A^{\prime}B^{\prime}}(P)=2$ if and only if $P$ is disconnected.*

Proof.

Assume $P$ is disconnected. Let $P_{1}$ be the vertices of a connected component of $P$ and let $P_{2}=V(P)-P_{1}$. Since, for all $u\in P_{1}$ and $v\in P_{2}$, $d(u,v)=\infty$ it follows that $\{P_{1},P_{2}\}$ satisfies conditions A*′* and B*′*.

Next, assume $P$ is connected but $\overline{P}=\{P_{1},P_{2}\}$ is a covering. Since $P$ is connected, $d(P_{1},P_{2})=1$ and there is a $p_{1}\in P_{1}$ and $p_{2}\in P_{2}$ with $d(p_{1},p_{2})=1$. Therefore $d(p_{1},P_{2})=1$ and $d(p_{2},P_{1})=1$ and condition B*′* fails, and so $\text{cov}_{A^{\prime}B^{\prime}}(P)>2$. ∎

Proposition 5.6**.**

For a graph $P$, $\text{cov}_{A^{\prime}}(P)=2$ if and only if $\text{diam}(P)\geq 5$.

Proof.

Assume $\text{diam}(P)\geq 5$. Then there exist $u,v\in V(P)$ with $d(u,v)\geq 5$. Let $P_{1}=\{p\in V(P):d(u,p)\leq 2\}$ and $P_{2}=V(P)-P_{1}$. Then $d(u,P_{2})\geq 3$. Also $d(v,P_{1})\geq 3$ because if there is a $p\in P_{1}$ with $d(v,p)\leq 2$, then $d(u,v)\leq d(u,p)+d(p,v)\leq 4$, a contradiction. So $\{P_{1},P_{2}\}$ satisfies condition A*′*.

Next, assume there is a covering $\{P_{1},P_{2}\}$ of $P$ satisfying condition A*′*. Then there is a $u\in P_{1}$ such that $d(u,P_{2})\geq 3$. Then $N_{2}[u]\subset P_{1}$ and $N_{2}[u]\cap P_{2}=\emptyset$. Similarly, $P_{2}$ contains a vertex $v$ such that $d(v,P_{1})\geq 3$, and so $N_{2}[v]\subset P_{1}$ and $N_{2}[v]\cap P_{1}=\emptyset$. Therefore $d(u,v)\geq 5$. ∎

Proposition 5.7**.**

If $P$ is a graph with $\text{diam}(P)\leq 3$, then $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)\neq 2$.

Proof.

Let $P$ be a graph with $\text{diam(P)}\leq 3$. Suppose there is a covering $\overline{P}=\{P_{1},P_{2}\}$ of $P$ satisfying condition A such that for $\iota=1$ the triple $(P,\overline{P},\overline{Q})$ satisfies conditions A*′′* and B*′′, where $\overline{Q}=\{Q_{0},Q_{1},P_{2}\}$ with $P_{1}=Q_{0}\cup Q_{1}$. For each $p\in P_{1}$, $d(p,P_{2})\geq 2$ by condition B′′*-2 because $\text{diam}(P)\leq 3$. This implies $d(P_{1},P_{2})\geq 2$. However, $d(P_{1},P_{2})\leq 1$ since $P$ is connected, a contradiction. ∎

Proposition 5.8**.**

If $P$ is a graph with $r(P)=2$, then $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)\neq 2$.

Proof.

Let $P$ be a graph with $r(P)=2$. Suppose there is a covering $\overline{P}=\{P_{1},P_{2}\}$ of $P$ satisfying condition A such that for $\iota=1$ the triple $(P,\overline{P},\overline{Q})$ satisfies conditions A*′′* and B*′′*, where $\overline{Q}=\{Q_{0},Q_{1},P_{2}\}$ with $P_{1}=Q_{0}\cup Q_{1}$.

For a central vertex $c\in V(P)$ either $c\in P_{1}$ or $c\in P_{2}$.

Suppose $c\in P_{1}$. Without loss of generality assume $c\in Q_{0}$. By condition B*′′*-2, $d(c,P_{2})\geq 2$. In fact, $d(c,P_{2})=2$ because $c$ is central, and so there exists a $p\in P_{2}$ with $d(c,p)=2$. Let $c-x-p$ be a dmpath. Then $x\in P_{1}$, $e(x)\leq 3$ and $d(x,P_{2})=1$. Therefore, $\{P_{1},P_{2}\}$ does not satisfy condition $B^{\prime\prime}$-2 for $x$, a contradiction.

Next, suppose $c\in P_{2}$. By the hypotheses $c$ does not satisfy B*′′-1a, B′′-1c, and B′′-1d. Therefore, $d(c,Q_{0})\geq 2$ and $d(c,Q_{1})\geq 2$. Since $c$ is a central vertex, $d(c,Q_{0})=2$ and $d(c,Q_{1})=2$. Then there exist $q_{0}\in Q_{0}$ and $q_{1}\in Q_{1}$, satisfying $d(c,q_{0})=2$ and $d(c,q_{1})=2$. Let $c-x-q_{0}$ and $c-y-q_{1}$ be paths. Then both $x$ and $y$ are in $P_{2}$ and $d(P_{2},Q_{i})\leq 1$ for $i=0,1$. Therefore $\{P_{1},P_{2}\}$ does not satisfy condition A′′*-1.

∎

To fully understand $A_{ucg}(C,P)$ there are still two cases left to consider, when $\text{diam}(P)=4$ and $r(P)=3$, and $\text{diam}(P)=4$ and $r(P)=4$. In these cases $\text{cov}_{A^{\prime}B^{\prime}}(P)\neq 2$ and $\text{cov}_{A^{\prime}}(P)\neq 2$. When $\text{diam}(P)=r(P)=4$, $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=2$ depends on $P$. To show that there exist graphs $P$ with $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)\neq 2$, we introduce the following lemma.

Lemma 5.9**.**

If $P$ is a graph with $\text{diam}(P)=4$ and $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=2$, then there exist three vertices $x_{1},x_{2},x_{3}$ such that

[TABLE]

Proof.

Suppose there is a covering $\overline{P}=\{P_{1},P_{2}\}$ of $P$ satisfying condition A such that for $\iota=1$ the triple $(P,\overline{P},\overline{Q})$ satisfies conditions A*′′* and B*′′*, where $\overline{Q}=\{Q_{0},Q_{1},P_{2}\}$ with $P_{1}=Q_{0}\cup Q_{1}$.

Since $P$ is connected without loss of generality we may assume that $d(P_{2},Q_{1})\leq 1$. Then A*′′-2b is not satisfied for $l=1$ and so, from A′′-2a, there exists a $p_{1}\in P_{2}$ such that $d(Q_{1},p_{1})\geq 3$. For $l=0$ there are two cases for condition A′′*-2, either part 2a is met or part 2b is met.

For $l=0$, either there exists a $p_{0}\in P_{2}$ such that $d(Q_{0},p_{0})\geq 3$ or $d(P_{2},Q_{0})\geq 2$ by condition A*′′*-2b.

Suppose $d(P_{2},Q_{0})\geq 2$, by connectivity of $P$ we know $d(P_{2},Q_{1})\leq 1$ and $d(Q_{0},Q_{1})\leq 1$. Since $d(p_{1},Q_{1})\geq 3$, $N_{2}[p_{1}]\cap Q_{1}=\emptyset$ and so

[TABLE]

Since $p_{1}\in P_{2}$ and $d(Q_{0},P_{2})\geq 2$, $N_{1}[p_{1}]\cap Q_{0}=\emptyset$ and $N_{1}[p_{1}]\subset P_{2}$. From

[TABLE]

it follows that $N_{2}[p_{1}]\subset P_{2}$. Therefore, $d(N_{2}[p_{1}],Q_{0})\geq 2$ and $d(p_{1},Q_{0})\geq 4$. Since $\text{diam}(P)=4$, $d(p_{1},Q_{0})=4$.

Let $p_{1}-x-y-z-q_{0}$ be a dmpath for some $q_{0}\in Q_{0}$. Then $z\not\in Q_{0}$ since $d(z,p_{1})=3$, and $z\not\in P_{2}$ since $d(q_{0},z)=1$. Hence $z\in Q_{1}$. Furthermore, $y\in N_{2}[p_{1}]\subset P_{2}$, so $d(z,P_{2})\leq 1$ and $d(z,Q_{0})\leq 1$. This implies condition B*′′*-2 is not met for $z\in Q_{1}$, a contradiction. Therefore $d(P_{2},Q_{0})\leq 1$.

Then, for $l=0$ condition A*′′*-2a is satisfied and there is a $p_{0}\in P_{2}$ such that $d(Q_{0},p_{0})\geq 3$. Therefore, $N_{2}[p_{0}]\cap Q_{0}=\emptyset$. Since $N_{2}[p_{1}]\cap Q_{1}=\emptyset$, it follows that $N_{2}[p_{0}]\cap N_{2}[p_{1}]\subset P_{2}$.

From condition A*′′*-1 there exists a $q\in Q_{0}$ such that $d(q,P_{2})\geq 3$. Then $N_{2}[q]\cap P_{2}=\emptyset$ and

[TABLE]

∎

Proposition 5.10**.**

There exists a graph $P$ with $r(P)=\text{diam}(P)=4$ and $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)\neq 2$.

Proof.

Let $P$ be the graph of a hexagonal prism as in figure 4. Then $r(P)=\text{diam}(P)=4$. If $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=2$ then by lemma 5.9 there exist vertices $x_{1},x_{2},x_{3}$ such that

[TABLE]

or equivalently,

[TABLE]

For any vertex $x$ of $P$, the compliment of $N_{2}[x]$ is a star with three pendant vertices as in figure 4. Because the prism has twelve vertices, a set of three stars would cover the prism with no overlap. However, one can check this is not possible and so $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)\neq 2$

∎

Proposition 5.11**.**

There exists a graph $P$ with $r(P)=\text{diam}(P)=4$ and $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=2$.

Proof.

Let $P$ be the graph of a heptagonal prism as in figure 5. Then $r(P)=\text{diam}(P)=4$. Consider the covering $\overline{P}=\{P_{1},P_{2}\}$, where $P_{1}=Q_{0}\cup Q_{1}$ as in figure 5. Here vertices of $Q_{0}$ are represented by the open circles, $Q_{1}$ by the open squares, and $P_{2}$ by the filled circles. One can verify this covering satisfies conditions A, A*′′* and B*′′*, and so $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=2$.

∎

6 Appendage Numbers

In this section we determine $A_{ucg}(C,P)$ for most pairs of graphs $(C,P)$ based on the structure of $C$. Conjectures are also given for the two remaining cases.

Theorem 6.1**.**

Let $C=\{v\}$ and $P$ be any graph with $r(P)\geq 2$. Then $A_{ucg}(C,P)=0$.

Proof.

Let $H$ be the graph with vertex set

[TABLE]

and edge set

[TABLE]

We claim $H$ is a UCG with $\langle\mathcal{Z}(H)\rangle=C$, $\langle\mathcal{CP}(H)\rangle=P$ and $|\mathcal{I}(H)|=0$.

Since $v$ is adjacent to all other vertices in $H$, $e(v)=1$. If $p$ is a vertex in $P$ there is a $p^{\prime}\in V(P)$ satisfying $d_{P}(p,p^{\prime})=2$. Since $p$ and $p^{\prime}$ are not adjacent in $P$, they are not adjacent in $H$ and $e(p)\geq 2$. This implies $\langle\mathcal{Z}(H)\rangle=\{v\}$ and $H$ is UCG with $\langle\mathcal{CP}(H)\rangle=P$. ∎

Theorem 6.2**.**

When $n\geq 2$

[TABLE]

Furthermore, for all $t\in\mathbb{N}$ there is a graph $P$ with $\text{diam}(P)=r(P)=2$ and $A_{ucg}(K_{n},P)\geq t$.

Proof.

This follows directly from proposition 3.2, theorem 3.3 and propositions 5.1, 5.2, 5.3, and 5.4. ∎

Conjecture 1**.**

If $\text{diam}(P)=r(P)=3$, then $A_{ucg}(K_{n},P)=2$.

When $\text{diam}(P)=r(P)=3$, $A_{ucg}(C,P)=2$ or $3$. However, all our examples show $A_{ucg}(K_{n},P)=2$, but we have not been able to prove this is always true.

Theorem 6.3**.**

If $C$ is a graph with $\text{diam}(C)>1$ then

[TABLE]

Furthermore, for all $t\in\mathbb{Z}$ there is a graph $P$ with $\text{diam}(P)=r(P)=2$ and $A_{ucg}(C,P)\geq t$.

Proof.

This follows directly from theorem 4.9, proposition 5.1 and propositions 5.4, through 5.11. ∎

The only case not accounted for is when $\text{diam}(P)=4$ and $r(P)=3$. In this case we know $\text{cov}_{A^{\prime}B^{\prime}}(P)\neq 2$, so $A_{ucg}(C,P)=5$ or $6$. We also know $\text{cov}_{A^{\prime}}(P)\neq 2$. Therefore $A_{ucg}(C,P)=5$ if and only if $\text{cov}_{AA^{\prime\prime}B^{\prime\prime}}(P)=2$. However, we have not found this to be the case for any such $P$. We also have been unable to show it is impossible, so we are left with the following conjecture.

Conjecture 2**.**

If $C$ is a graph with $\text{diam}(C)>1$ and $P$ is a graph with $\text{diam}(P)=4$ and $r(P)=3$, then $A_{ucg}(C,P)=6$.

The $\text{diam}(P)=r(P)=4$ case also warrants further discussion. Propositions 5.10 and 5.11 show there are examples of $P$ when $A_{ucg}(C,P)=6$ and with $A_{ucg}(C,P)=5$. It will be necessary to find another metric invariant other than diameter and radius to refine the results of this case. At this point we are unsure what a suitable invariant may be.

Finally, independent of $C$, there is a major difference between possible appendage numbers when $\text{diam}(P)>2$ and when $\text{diam}(P)=2$. When $\text{diam}(P)>2$, theorems 6.2 and 6.3 show there are only finitely many possible appendage numbers, and which are independent of the size of $V(P)$. On the other hand, for $\text{diam}(P)=r(P)=2$ the graph $P^{\alpha}$ in proposition 5.4 gives

[TABLE]

This may suggest that appendage numbers are related to $|V(P)|$ when $\text{diam}(P)=r(P)=2$, however the following proposition shows this is not the case.

Proposition 6.4**.**

For every $\alpha,\beta\in\mathbb{N}$ there is a graph $P$ such that $V(P)=2\alpha+\beta$ and $A_{ucg}(K_{n},P)=2\alpha$.

Proof.

In this proof we modify the construction of $P^{\alpha}$ from proposition 5.4. For $\alpha,\beta\in\mathbb{N}$ define $P^{\alpha,\beta}=(V^{\alpha,\beta},E^{\alpha,\beta})$ as follows. Let

[TABLE]

and

[TABLE]

That is, $P^{\alpha,\beta}$ is the graph from proposition 5.4 with $\beta$ new vertices, $g_{i}^{\prime}s$, that are adjacent to every vertex except themselves, $e_{1}$ and $f_{1}$. Observe that $\text{diam}(P^{\alpha,\beta})=r(P^{\alpha,\beta})=2$ if $\alpha\geq 2$.

We now show $\text{cov}_{A}(P^{\alpha,\beta})=\text{cov}_{AB}(P^{\alpha,\beta})=2\alpha$. Let $\overline{P^{\alpha,\beta}}=\{P_{1},\ldots,P_{k}\}$ be a covering of $P^{\alpha,\beta}$ satisfying condition A. For $2\leq i\leq\alpha$, $\{e_{i}\},\{f_{i}\}\in\overline{P^{\alpha,\beta}}$ as in proposition 5.4. If

[TABLE]

then $\overline{P^{\alpha,\beta}}$ fails to satisfy condition A. Therefore $e_{1}$, $f_{1}$ and the $g_{l}$s must be contained in least two elements of $\overline{P^{\alpha,\beta}}$, and so $\text{cov}_{A}(P^{\alpha,\beta})\geq 2\alpha$.

Let $P^{\alpha,\beta}_{1}=\{e_{1}\}$ and $Q^{\alpha,\beta}_{1}=\{f_{1},g_{1}\ldots,g_{\beta}\}$, and for $2\leq i\leq\alpha$ let $P^{\alpha,\beta}_{i}=\{e_{i}\}$ and $Q^{\alpha,\beta}_{i}=\{f_{i}\}$. Then $\{P^{\alpha,\beta}_{1},Q^{\alpha,\beta}_{1},\ldots,P^{\alpha,\beta}_{\alpha},Q^{\alpha,\beta}_{\alpha}\}$ is a covering of $P^{\alpha,\beta}$ that satisfies both conditions A and B. So $A_{ucg}(K_{n},P^{\alpha,\beta})=2\alpha$ and $V(P^{\alpha,\beta})=2\alpha+\beta$.

∎

7 Other Appendage Numbers

In the paper [3] Gu defines $A_{ucg}(C)$ to be the minimum number of vertices needed to be added to $C$ in order to create a uniform central graph $G$ with $\langle\mathcal{Z}(G)\rangle=C$. To match notation we let $A_{ucg}(C,-)=A_{ucg}(C)$. Gu’s main theorem is the following.

Theorem 7.1** (Gu).**

If $C$ is a connected graph, then

[TABLE]

We use our results to give an alternative proof of Gu’s result which is also true without the condition that $C$ is connected.

Proof.

Observe that $A_{ucg}(C,-)=\text{min}\{A_{ucg}(C,P)+|V(P)|\}$ where the minimum is taken over all graphs $P$. By proposition 2.4 $r(P)>1$ and so we may assume $|V(P)|\geq 2$. Since $\text{cov}_{A}(P)\geq 2$ for any $P$, by proposition 3.2 $A_{ucg}(K_{n},P)\geq 2$ for $n\geq 2$, and by proposition 4.1 $A_{ucg}(C,P)\geq 4$ for any non-complete graph $C$. Hence $A_{ucg}(K_{n},-)\geq 4$ and $A_{ucg}(C,-)\geq 6$.

Let $P^{2}$ be a graph with two isolated vertices $\{u,v\}$ and $\overline{P^{2}}=\{\{u\},\{v\}\}$ a covering. Note $\overline{P^{2}}$ satisfies conditions A, B, A*′* and B*′*. Let

[TABLE]

and

[TABLE]

The graph $G_{1}$ is the UCG in the first half of the proof of theorem 3.3 and $G_{2}$ is a UCG by proposition 4.2. Furthermore, $|\mathcal{I}(G_{1})|=2$ and $|\mathcal{I}(G_{1})|=4$, so $A_{ucg}(K_{n},-)=4$ and $A_{ucg}(C,-)=6$.

Finally, $A_{ucg}(\{v\},-)=2$ by theorem 6.1. ∎

We can also consider $A_{ucg}(-,P)$, the minimum number of vertices needed to be added to $P$ in order to construct a uniform central graph $G$ with $\langle\mathcal{CP}(G)\rangle=P$. From propositions 2.4 and 6.1, it follows that $A_{ucg}(-,P)=\infty$ if $r(P)=1$, and $A_{ucg}(-,P)=1$ otherwise.