On the $g$-good-neighbor connectivity of graphs

TL;DR

This paper investigates the properties and bounds of the $g$-good-neighbor connectivity in graphs, characterizes specific cases, and establishes extremal results to enhance understanding of fault tolerance in interconnection networks.

Contribution

It provides new bounds, characterizations, and extremal results for the $g$-good-neighbor connectivity, advancing the theoretical understanding of fault tolerance parameters.

Findings

Bounds: $1 \,\leq\, \kappa^g(G) \leq n-2g-2$ for certain $g$.

Characterizations of graphs with $\,\kappa^g(G)=1,2$ and trees with $\,\kappa^g(T_n)=n-t$.

Extremal results for the $g$-good-neighbor connectivity.

Abstract

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq \kappa^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{\Delta(G),\left\lfloor \frac{n-3}{2}\right\rfloor\right\}$, and graphs with $\kappa^g(G)=1,2$ and trees with $\kappa^g(T_n)=n-t$ for $4\leq t\leq \frac{n+2}{2}$ are characterized, respectively. In the end, we get the three extremal results for the $g$-good-neighbor connectivity.

Equations

$$0\le g\le \left\{\Delta (G),\lfloor \frac{n-3}{2}\rfloor \right\}$$

$$e(G)\le \left(\genfrac{}{}{0ex}{}{n}{2}\right)-(g+1{)}^2.$$

$$2(g+1)\le |V(C_1)|+|V(C_2)|\le \sum _{i=1}^r|V(C_i)|=|V(G)|-|X|\le n-1,$$

$${\kappa }^g(G)\le {\kappa }^{g+1}(G).$$

$$\kappa (G)\le {\kappa }^g(G)\le n-2g-2.$$

$$1\le {\kappa }^g(G)\le n-2g-2.$$

$${\kappa }^0(K_{n_1,n_2,\dots ,n_r})=\sum _{i=1}^{r-1}{n}_i$$

$${\kappa }^1(T_n^{\ast })=n-t.$$

$$\{w_{j_1},w_{j_2},\dots ,w_{j_r}\}\cup \left(\bigcup _{i=1}^{r}V(K_{1,a_{j_i}-1})\right)=\{w_1,w_2,\dots ,w_r\}\cup \left(\bigcup _{i=1}^{r}V(K_{1,a_i-1})\right)\subseteq X.$$

$${\kappa }^1(T_n)\le |\{v_j\}\cup W|\le n-(n-t)-2-2\le t-4<n-t,$$

$${\kappa }^g(F_n^k)=k.$$

$${\kappa }^g(H_n^k)=k.$$

$${\kappa }^1(T_n^{\prime })=k.$$

$$s(n,k)=\frac{(n-k)g}{2}+k+1.$$

$$s(n,k)=\frac{(n-k)g+1}{2}+k+1.$$

$${\kappa }^g(G_n^k)=k-1.$$

$$f(n,k)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)-(n-k-g)(g+1)+1.$$

Full text

Keywords: Connectivity, $g$-good-neighbor connectivity, extremal problem.

AMS subject classification 2010: 05C40; 05C05; 05C76.

1 Introduction

With the rapid development of VLSI technology, a multiprocessor system may contain hundreds or even thousands of nodes, and some of them may be faulty when the system is implemented. As the number of processors in a system increases, the possibility that its processors may be comefaulty also increases. Because designing such systems without defects is nearly impossible, reliability and fault tolerance are two of the most critical concerns of multiprocessor systems [37].

By the definition proposed by Esfahanian [6], a multiprocessor system is fault tolerant if it can remain functional in the presence of failures. Two basic functionality criteria have received considerable attention. The first criterion for a system to be regarded as functional is whether the network logically contains a certain topological structure. This is the problem that occurs when embedding one architecture into another [15, 35]. This approach involves using system-wide redundancy and reconfiguration. The second functionality criterion considers a multiprocessor system functional if a fault-free communication path exists between any two fault-free nodes; that is, the topological structure of the multiprocessor system remains connected in the presence of certain failures. Thus, connectivity and edge connectivity are two major measurements of this criterion [35]. The connectivity of a graph $G$, denoted by $\kappa(G)$, is the minimal number of vertices whose removal from produces a disconnected graph or only one vertex; the edge connectivity of a graph $G$, denoted by $\lambda(G)$, is the minimal number of edges whose removal from produces a disconnected graph. However, these two parameters tacitly assume that all vertices that are adjacent to, or all edges that are incident to, the same vertex can potentially fail simultaneously. This is practically impossible in some network applications.

For a graph $G$, let $V(G)$, $E(G)$, $e(G)$, $\overline{G}$, and $diam(G)$ denote the set of vertices, the set of edges, the size, the complement, and the diameter of $G$, respectively. A subgraph $H$ of $G$ is a graph with $V(H)\subseteq V(G)$, $E(H)\subseteq E(G)$, and the endpoints of every edge in $E(H)$ belonging to $V(H)$. For any subset $X$ of $V(G)$, let $G[X]$ denote the subgraph induced by $X$; similarly, for any subset $F$ of $E(G)$, let $G[F]$ denote the subgraph induced by $F$. We use $G-X$ to denote the subgraph of $G$ obtained by removing all the vertices of $X$ together with the edges incident with them from $G$; similarly, we use $G-F$ to denote the subgraph of $G$ obtained by removing all the edges of $F$ from $G$. If $X=\{v\}$ and $F=\{e\}$, we simply write $G-v$ and $G-e$ for $G-\{v\}$ and $G-\{e\}$, respectively. For two subsets $X$ and $Y$ of $V(G)$ we denote by $E_{G}[X,Y]$ the set of edges of $G$ with one end in $X$ and the other end in $Y$. If $X=\{x\}$, we simply write $E_{G}[x,Y]$ for $E_{G}[\{x\},Y]$. The degree of a vertex $v$ in a graph $G$, denoted by $deg_{G}(v)$, is the number of edges of $G$ incident with $v$. Let $\delta(G)$ and $\Delta(G)$ be the minimum degree and maximum degree of the vertices of $G$, respectively. The set of neighbors of a vertex $v$ in a graph $G$ is denoted by $N_{G}(v)$. The union $G\cup H$ of two graphs $G$ and $H$ is the graph with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)$. If $G$ is the disjoint union of $k$ copies of a graph $H$, we simply write $G=kH$.

1.1 The $g$-extra (edge-)connectivity

Fàbrega and Fiol [8, 9] introduced the following measures. A subset of vertices $S$ is said to be a cutset if $G-S$ is not connected. A cutset $S$ is called an $R_{g}$-cutset, where $g$ is a non-negative integer, if every component of $G-S$ has at least $g+1$ vertices. If $G$ has at least one $R_{g}$-cutset, the $g$-extra connectivity of $G$, denoted by $\kappa_{g}(G)$, is then defined as the minimum cardinality over all $R_{g}$-cutsets of $G$. A connected graph $G$ is said to be $g$-extra connected if $G$ has a $g$-extra cut.

Given a graph and a non-negative integer $g$, the $g$-extra edge-connectivity, written as $\lambda_{g}(G)$, is the minimal cardinality of a set of edges in $G$, if it exists, whose deletion disconnects $G$ and leaves each remaining component with more than $g$ vertices.

Note that $\kappa_{0}(G)=\kappa(G)$ and $\lambda_{0}(G)=\lambda(G)$ for any graph if is not a complete graph. Therefore, the $g$-extra connectivity (resp. $g$-extra edge connectivity) can be regarded as a generalization of the classical connectivity (resp. classical edge connectivity) that provides measures that are more accurate for reliability and fault tolerance for large-scale parallel processing systems. Regarding the computational complexity of the problem, based on thorough research, no polynomial-time algorithm has been presented to compute $\kappa_{g}$ for a general graph; nor has there been any tight upper bound for $\kappa_{g}$ [6]. However, $\lambda_{1}$ can be computed by solving numerous network flow problems [7].

1.2 The $g$-good-neighbor connectivity

Let $G=(V,E)$ be connected. A fault set $F\subseteq V$ is called a $g$-good-neighbor faulty set if $|N(v)\cap(V-F)|\geq g$ for every vertex $v$ in $V-F$. A $g$-good-neighbor cut of $G$ is a $g$-good-neighbor faulty set $F$ such that $G-F$ is disconnected. The minimum cardinality of $g$-good-neighbor cuts is said to be the $g$-good-neighbor connectivity of $G$, denoted by $\kappa^{g}(G)$. A connected graph $G$ is said to be $g$-good-neighbor connected if $G$ has a $g$-good-neighbor cut. For more research on $g$-good-neighbor connectivity, we refer to [16, 18, 22, 24, 28, 29, 33, 32, 36, 38, 39].

The relation of $g$-extra connectivity and $g$-good-neighbor connectivity was given in [23].

Theorem 1.1**.**

[23]* $(1)$ If $G$ is a $1$-good-neighbor connected graph, then $\kappa_{1}(G)=\kappa^{1}(G)$.*

$(2)$* If $G$ is a $g$-extra and $g$-good-neighbor connected graph, then $\kappa_{g}(G)\leq\kappa^{g}(G)$.*

The range of the integer $g$ can be determined immediately.

Proposition 1.1**.**

Let $g$ be a non-negative integer. If $G$ has its $g$-good-neighbor connectivity, then

[TABLE]

and

[TABLE]

Proof.

From the definition of $g$-good-neighbor connectivity, there exists $X\subseteq V(G)$ with $|X|=\kappa_{g}(G)$ such that $G-X$ is not connected and the minimum degree of each component of $G-X$ is at least $g$. Let $C_{1},C_{2},\ldots,C_{r}$ be the components of $G-X$. Then $r\geq 2$ and

[TABLE]

and hence $0\leq g\leq\lfloor\frac{n-3}{2}\rfloor$. For each $C_{i}\ (1\leq i\leq r)$, we have $\Delta(G)\geq\Delta(C_{i})\geq\delta(C_{i})\geq g$.

Furthermore, $e(\overline{G})\geq|V(C_{1})||V(C_{2})|=(g+1)^{2}$, and hence $e(G)\leq{n\choose 2}-(g+1)^{2}$. ∎

The monotone property of $\kappa_{g}(G)$ for non-negative integer $g$ is true.

Proposition 1.2**.**

Let $g$ be a non-negative integer, and let $G$ be a connected graph. Then

[TABLE]

Proof.

From the definition of $(g+1)$-good-neighbor connectivity, by deleting $\kappa^{g+1}(G)$ vertices in $G$, the resulting graph is not connected and the minimum degree of each component is at least $g+1>g$, and hence $\kappa^{g}(G)\leq\kappa^{g+1}(G)$. ∎

The monotone property of $\kappa^{0}(G)$ is true in terms of connected graphs $G$.

Observation 1.1**.**

Let $G$ be a connected graph. If $H$ is a spanning subgraph of $G$, then $\kappa^{0}(H)\leq\kappa^{0}(G)$.

But for $g\geq 1$, the above monotone property is not true.

Remark 1.1**.**

Let $G$ be a graph obtained from four cliques $X_{1},X_{2},Y_{1},Y_{2}$ with $\delta(X_{i})\geq g\ (i=1,2)$ and $\delta(Y_{j})\geq g\ (j=1,2)$ and three vertices $u,v,w$ by adding edges in $E_{G}[u,X_{1}]\cup E_{G}[u,X_{2}]\cup E_{G}[u,Y_{1}]\cup E_{G}[u,Y_{2}]\cup E_{G}[v,Y_{1}]\cup E_{G}[v,Y_{2}]\cup E_{G}[w,Y_{1}]\cup E_{G}[w,Y_{2}]\cup\{uv,uw\}$. Let $H$ be a graph obtained from two cliques $Y_{1},Y_{2}$ defined in $G$, two subgraph $Z_{1},Z_{2}$ such that $\delta(Z_{i})=g-1\ (i=1,2)$, and three vertices $u,v,w$ by adding edges in $E_{G}[u,Z_{1}]\cup E_{G}[u,Z_{2}]\cup E_{G}[u,Y_{2}]\cup E_{G}[v,Y_{1}]\cup E_{G}[v,Y_{2}]\cup E_{G}[w,Y_{1}]\cup E_{G}[w,Y_{2}]$. Clearly, $H$ is a spanning subgraph of $G$; see Figure $1$. We first show that $\kappa^{g}(G)=1$. By deleting the vertex $u$, there are three components and the minimum degree of each component is at least $g$, and hence $\kappa^{g}(G)\leq 1$, and so $\kappa^{g}(G)=1$. Next, we show that $\kappa^{g}(H)=2$. Suppose $\kappa^{g}(H)=1$. Then there exists a vertex such that by deleting this vertex the resulting graph is not connected and the minimum degree of each component is at least $g$. Note that $u$ is the unique cut vertex of $H$. Clearly, there is a component having minimum degree $g-1$, a contradiction. So $\kappa^{g}(H)\geq 2$. By deleting the vertices $v,w$, there are two components and the minimum degree of each of them is at least $g$, and hence $\kappa^{g}(H)\leq 2$. So $\kappa^{g}(H)=2>\kappa^{g}(G)$.

1.3 Some classical problems

One of the interesting problems in extremal graph theory is the Erdös-Gallai-type problem, which is to determine the maximum or minimum value of a graph parameter with some given properties. In [1, 12], the authors investigated two kinds of Erdös-Gallai-type problems for monochromatic connection number and monochromatic vertex connection number, respectively. Motivated by these, we study two kinds of Erdös-Gallai-type problems for $\kappa^{g}(G)$ in this paper.

Problem 1. Given two positive integers $n$ and $k$, compute the minimum integer $f(n,k)$ such that for every connected graph $G$ of order $n$, if $|E(G)|\geq f(n,k)$ then $\kappa^{g}(G)\geq k$.

Problem 2. Given two positive integers $n$ and $k$, compute the maximum integer $g(n,k)$ such that for every graph $G$ of order $n$, if $|E(G)|\leq g(n,k)$ then $\kappa^{g}(G)\leq k$.

Another interesting problem in extremal graph theory is to study the minimum size of graphs with given parameter; see [25].

Problem 3. Given two positive integers $n$ and $k$, compute the minimum integer $s(n,k)=\min\{|E(G)|:G\in\mathscr{G}(n,k)\}$, where $\mathscr{G}(n,k)$ the set of all graphs of order $n$ (that is, with $n$ vertices) with $g$-good-neighbor connectivity $k$.

In Section $2$, we obtain the exact values of $g$-extra connectivities of complete bipartite graphs, complete multipartite graphs, wheels and paths. For a connected graph $G$ of order $n$, we show that $\kappa(G)\leq\kappa^{g}(G)\leq n-2g-2$ for $0\leq g\leq\min\{\Delta(G),\left\lfloor\frac{n-3}{2}\right\rfloor\}$, and $1\leq\kappa^{g}(G)\leq n-2g-2$ for $0\leq g\leq\left\lfloor\frac{n-3}{2}\right\rfloor$ in Section $2$. Graphs with $\kappa_{g}(G)=1,2$ and trees with $\kappa^{g}(T_{n})=n-t$ are characterized, respectively, in Section $3$. In the end, we get the extremal results for the $g$-good neighbor connectivity in Section $4$.

2 Results for special graphs

The following upper and lower bounds are immediate.

Proposition 2.1**.**

Let $G$ be a connected graph of order $n$, and let $g$ be a non-negative integer such that $0\leq g\leq\min\{\Delta(G),\left\lfloor\frac{n-3}{2}\right\rfloor\}$. Then

[TABLE]

Moreover, the upper and lower bounds are sharp.

Proof.

From the definition of $\kappa^{g}(G)$, we have $\kappa^{g}(G)\geq\kappa(G)$. Suppose $\kappa^{g}(G)\geq n-2g-1$. From the definition, we can delete $\kappa^{g}(G)$ vertices in $G$ such that there are at least two components and one of them has no more than $g$ vertices, a contradiction. So $\kappa(G)\leq\kappa^{g}(G)\leq n-2g-2$. Theorem 3.1 shows that the upper bound is sharp. If $k=0$, then $\kappa(G)=\kappa^{0}(G)$. This implies that the lower bound is sharp. ∎

The following corollary is immediate from Proposition 2.1.

Corollary 2.1**.**

Let $n,g$ be two integers with $0\leq g\leq\left\lfloor\frac{n-3}{2}\right\rfloor$. If $G$ is a connected graph of order $n$, then

[TABLE]

Moreover, the upper and lower bounds are sharp.

In the following, we obtain the exact values for $g$-good neighbor connectivity of some special graphs.

Proposition 2.2**.**

Let $g$ be a non-negative integer.

$(1)$* If $K_{a,b}\ (a\geq b\geq 2)$ is a complete bipartite graph, then $g=0$ and $\kappa^{0}(K_{a,b})=b$ and $\kappa^{g}(K_{a,b})$ does not exist for $g\geq 1$.*

$(2)$* Let $r$ be an integer with $r\geq 3$. For complete multipartite graph $K_{n_{1},n_{2},\ldots,n_{r}}$ $(n_{1}\leq n_{2}\leq\ldots\leq n_{r})$, we have $g=0$ and*

[TABLE]

and $\kappa^{g}(K_{n_{1},n_{2},\ldots,n_{r}})$ does not exist for $g\geq 1$.

Proof.

$(1)$ By deleting any vertex in $K_{a,b}$, the resulting graph is still a complete bipartite graph and it is connected. If we require the resulting graph is not connected, then we must delete all the vertices of one part. Then $g=0$. Since $a\geq b\geq 2$, we have $\kappa^{g}(K_{a,b})=b$.

$(2)$ Similarly to the proof of $(1)$, we can get $\kappa^{g}(K_{n_{1},n_{2},\ldots,n_{r}})=\sum_{i=1}^{r-1}n_{i}$. ∎

Proposition 2.3**.**

Let $g$ be a non-negative integer.

$(1)$* If $W_{n}\ (n\geq 7)$ is a wheel of order $n$, then $\kappa^{g}(W_{n})=3$ for $g=0,1$.*

$(2)$* If $P_{n}\ (n\geq 3)$ be a path of order $n$, then $\kappa^{g}(P_{n})=1$ for $g=0,1$.*

Proof.

$(1)$ From the definition of $\kappa^{g}(W_{n})$, there exists $X\subseteq V(W_{n})$ such that $W_{n}-X$ is not connected and the minimum degree of each component of $W_{n}-X$ is at least $g$. Note that each component is a path. Then $g\leq 1$. From Proposition 2.1, we have $\kappa^{g}(W_{n})\geq\kappa(W_{n})=3$. It suffices to show that $\kappa^{g}(W_{n})\leq 3$ for $g=0,1$. Let $v$ be the center of $W_{n}$, and $W_{n}-v=C_{n-1}$, and $V(C_{n-1})=\{u_{1},u_{2},\ldots,u_{n-1}\}$. Choose $X=\{v,u_{1},u_{\lceil\frac{n}{2}\rceil}\}$. Since $g=0,1$, it follows that the minimum degree of each component of $W_{n}-X$ is at least $g$, and hence $\kappa^{g}(W_{n})\leq 3$. So $\kappa^{g}(W_{n})=3$.

$(2)$ Similarly to the proof of $(1)$, we have $g=0,1$. From Proposition 2.1, we have $\kappa^{g}(P_{n})\geq\kappa(P_{n})=1$. It suffices to show $\kappa^{g}(P_{n})\leq 1$. Let $P_{n}=u_{1}u_{2}\ldots u_{n}$. Choose $v=u_{\lceil n/2\rceil}$. Since $0\leq g\leq\lfloor\frac{n-1}{2}\rfloor-1$, it follows that the minimum degree of each component of $G-v$ is at least $g$, and hence $\kappa^{g}(P_{n})\leq 1$. So $\kappa^{g}(P_{n})=1$. ∎

3 Graphs with given $g$-good-neighbor connectivity

In this section, we first characterize trees with given $g$-good-neighbor connectivity. Next, we characterize graphs with small $g$-good-neighbor connectivity.

3.1 Trees with given $g$-good-neighbor connectivity

Let $K_{1,n-t-1}$ be a star with center $v$ and leaves $u_{1},u_{2},\ldots,u_{n-t-1}$, and let $K_{1,a_{1}-1},K_{1,a_{2}-1},\ldots,K_{1,a_{r}-1}$ be $r$ stars with centers $w_{1},w_{2},\ldots,w_{r}$, respectively. Furthermore, let $T_{n}^{*}$ be a tree of order $n$ obtained from $K_{1,n-t-1}$ and $K_{1,a_{1}-1},K_{1,a_{2}-1},\ldots,K_{1,a_{r}-1}$ by adding the edges $\{vw_{1},vw_{2},\ldots,vw_{r}\}$, where $r\geq 2$, $\sum_{i=1}^{r}a_{i}=t$, and $a_{i}\geq 2$ for each $i\ (1\leq i\leq r)$; see Figure 1.

Lemma 3.1**.**

For $4\leq t\leq\frac{n+2}{2}$, we have

[TABLE]

Proof.

Choose $X=\{u_{i}\,|\,1\leq i\leq n-t-1\}\cup\{v\}$. Then $T_{n}^{*}-X$ is not connected and the minimum degree of each component is exactly $1$. So $\kappa^{1}(T_{n}^{*})\leq|X|=n-t$. It suffices to show $\kappa^{1}(T_{n}^{*})\geq n-t$. It suffices to prove that for any $X\subseteq V(T_{n}^{*})$ and $|X|\leq n-t-1$, if $T_{n}^{*}-X$ is not connected, then there exists a component of $T_{n}^{*}-X$ such that its minimum degree is exactly [math]. If $v\in X$, then there is an isolated vertex in the resulting graph, as desired.

Suppose $v\notin X$. Since $T_{n}^{*}-X$ is not connected, it follows that there exits some $w_{j_{1}}$ such that $w_{j_{1}}\in X$, and hence $V(K_{1,a_{j_{1}}-1})\subseteq X$. Then there exits some $w_{j_{2}}$ such that $w_{j_{2}}\in X-w_{j_{1}}$, and hence $V(K_{1,a_{j_{2}}-1})\subseteq X$. Furthermore, there exits some $w_{j_{3}}$ such that $w_{j_{3}}\in X-w_{j_{1}}-w_{j_{2}}$, and hence $V(K_{1,a_{j_{3}}-1})\subseteq X$. Continue this process, we have

[TABLE]

Clearly, $T_{n}^{*}-X$ is connected, a contradiction. So $\kappa^{1}(T_{n}^{*})\geq n-t$, and hence $\kappa^{1}(T_{n}^{*})=n-t$. ∎

Trees with $\kappa^{g}(T_{n})=n-t$ for general $g$ and $t$ can be characterized.

Theorem 3.1**.**

Let $n,g$ be two integers and $T_{n}$ be a tree of order $n$ with $0\leq g\leq\left\{\Delta(T_{n}),\left\lfloor\frac{n-3}{2}\right\rfloor\right\}$. Then $\kappa^{g}(T_{n})=n-t$ if and only if $T_{n}$ satisfies one of the following conditions.

$(1)$* $g=0$ and $n=t+1$;*

$(2)$* $g=1$, $4\leq t\leq\frac{n+2}{2}$ and $T_{n}=T_{n}^{*}$.*

Proof.

If $g=0$ and $n=t+1$, then $\kappa^{0}(T_{t+1})=1=(t+1)-t=n-t$. Suppose $g=1$, $4\leq t\leq\frac{n+2}{2}$ and $T_{n}=T_{n}^{*}$. From Lemma 4.1, we have $\kappa^{1}(T_{n}^{*})=n-t$.

Conversely, we suppose $\kappa^{g}(T_{n})=n-t$. Then we have the following claim.

Claim 1**.**

$g\leq 1$.

Proof.

Assume, to the contrary, that $g\geq 2$. From the definition of $\kappa^{g}(T_{n})$, there exists $X\subseteq V(T_{n})$ and $|X|=\kappa^{g}(T_{n})$ such that $T_{n}-X$ is not connected and the minimum degree of each component of $T_{n}-X$ is at least $g\geq 2$. Since $T_{n}$ is a tree, it follows that each component of $T_{n}-X$ is a subtree of $T_{n}$, and the minimum degree of each component is at most $1$, a contradiction. ∎

From Claim 1, we have $g\leq 1$. If $g=0$, then $1=\kappa(T_{n})=\kappa^{0}(T_{n})=n-t$, and hence $n=t+1$. If $g=1$, then $\kappa^{1}(T_{n})=n-t$. Then there exists $X\subseteq V(T_{n})$ and $|X|=n-t$ such that $T_{n}-X$ is not connected and the minimum degree of each component is exactly $1$. Clearly, there exits a cut vertex $v$ in $T_{n}$ such that $v\in X$. Let $C_{1},C_{2},\ldots,C_{r}$ be the components of $T_{n}-v$. Since $T_{n}$ is a tree, it follows that $|E_{G}[v,V(C_{i})]|=1$ for each $i\ (1\leq i\leq r)$. Let $x$ be the number of isolated vertices in $T_{n}-v$. Since $|X|=n-t$ and $v\in X$, it follows that $x\leq n-t-1$. Furthermore, we have the following claim.

Claim 2**.**

$x=n-t-1$.

Proof.

Assume, to the contrary, that $x\leq n-t-2$. By deleting these $x$ isolated vertices and $v$, the minimum degree of each component of the resulting graph is at least $1$, and hence $\kappa^{1}(T_{n})\leq x+1\leq n-t-2+1=n-t-1<n-t$, a contradiction. ∎

Let $C_{i}\ (1\leq i\leq n-t-1)$ be the isolated vertices in $T_{n}-v$. Then we have the following claim.

Claim 3**.**

For each $i\ (n-t\leq i\leq r)$, $C_{i}$ is a star.

Proof.

Assume, to the contrary, that there exists some $C_{j}$ such that $C_{j}$ is not a star. Then $C_{j}$ contains a $2K_{2}$, say $u_{1}u_{2},u_{3}u_{4}$. Let $vv_{j}$ be the unique edge from $v$ to $C_{j}$. Let $W$ be the set of pendent vertices adjacent to $v_{j}$ in $C_{j}$. Since $C_{j}$ is not a star, it follows that there is at least one edge in $C_{j}-v_{j}-W$. Since $T_{n}-X$ is not connected, it follows that except $C_{j}$, there exists another component of order at least $2$, say $C_{k}$. Then

[TABLE]

a contradiction. ∎

From Claim 3, $C_{i}$ is a star for each $i\ (n-t\leq i\leq r)$. Let $w_{i}$ be the center of $C_{i}$, where $n-t\leq i\leq r$. Then we have the following claim.

Claim 4**.**

For each $i\ (n-t\leq i\leq r)$, we have $vw_{i}\in E(T_{n})$.

Proof.

Assume, to the contrary, that there exists some $w_{j}$ such that $vw_{j}\notin E(T_{n})$. Then there exists a vertex $a_{j}$ such that $a_{j}v\in E(T_{n})$. Note that $a_{j}$ is a leaf of $C_{j}$. Then $T_{n}-a_{j}$ is not connected and the minimum degree of each component is at least $1$, and hence $\kappa^{1}(T_{n})\leq 1$, a contradiction. ∎

From Claim 4, $vw_{i}\in E(T_{n})$ for each $i\ (n-t\leq i\leq r)$. Then $T_{n}=T_{n}^{*}$. ∎

3.2 Graphs with small $g$-good-neighbor connectivity

Graphs with $\kappa^{g}(G)=1$ can be characterized easily.

Observation 3.1**.**

Let $n,g$ be two integers and let $G$ be a connected graph of order $n$ with $0\leq g\leq\left\{\Delta(T_{n}),\left\lfloor\frac{n-3}{2}\right\rfloor\right\}$. Then $\kappa^{g}(G)=1$ if and only if there exists a cut vertex $v$ in $G$ such that the minimum degree of each connected component of $G-v$ is at least $g$.

We can also characterize graphs with $\kappa^{g}(G)=2$.

Theorem 3.2**.**

Let $n,g$ be two integers and let $G$ be a connected graph of order $n$ with $0\leq g\leq\left\{\Delta(T_{n}),\left\lfloor\frac{n-3}{2}\right\rfloor\right\}$. Then $\kappa^{g}(G)=2$ if and only if $G$ satisfies one of the following conditions.

$(1)$* $\kappa(G)=2$ and there exists a cut vertex set $\{u,v\}$ in $G$ such that the minimum degree of each component of $G-\{u,v\}$ is at least $g$;*

$(2)$* $\kappa(G)=1$, and $g\geq 1$, and $(2.1),(2.2)$ hold, where*

• $(2.1)$* for each cut vertex $u$, there exists a component of $G-u$ such that its minimum degree is at most $g-1$,*
• $(2.2)$* $(a)$ there exists a cut vertex $v$ such that there is exactly one component in $G-v$ having exactly one vertex $u$ of degree at most $g-1$ and the neighbors of $u$ has degree at least $g+1$ and the minimum degree of other vertices is at least $g$, or $(b)$ there exists a cut vertex $v$ such that $G-v$ contains at least $3$ components, where one of the component is an isolated vertex and the minimum degree of each of the other components is at least $g$, or $(c)$ there are two non-cut vertices $x,y$ such that $G-\{x,y\}$ is not connected and the minimum degree of each component is at least $g$.*

Proof.

Suppose that $G$ satisfies $(1)$ and $(2)$. Suppose that $(1)$ holds. Since the minimum degree of each component of $G-\{u,v\}$ is at least $g$, it follows that $\kappa^{g}(G)\leq 2$. From Proposition 2.1, we have $\kappa^{g}(G)\geq\kappa(G)=2$.

Suppose that $(2)$ holds. Since for each cut vertex $u$, there exists a component of $G-u$ such that its minimum degree is at most $g-1$, it follows that $\kappa^{g}(G)\geq 2$. If there exists a cut vertex $v$ such that there is exactly one component in $G-v$ having exactly one vertex $u$ of degree at most $g-1$ and the neighbors of $u$ has degree at least $g+1$ and the minimum degree of other vertices is at least $g$, then $\kappa^{g}(G)\leq|\{u,v\}|=2$. If there exists a cut vertex $v$ such that $G-v$ contains at least $3$ components, where one of the component is an isolated vertex $u$ and the minimum degree of each of the other components is at least $g$, then $\kappa^{g}(G)\leq|\{u,v\}|=2$. If there are two non-cut vertices $x,y$ such that $G-\{x,y\}$ is not connected and the minimum degree of each component is at least $g$, $\kappa^{g}(G)\leq|\{x,y\}|=2$. So we have $\kappa^{g}(G)=2$.

Conversely, we suppose $\kappa^{g}(G)=2$. From Proposition 2.1, we have $\kappa(G)\leq 2$. Suppose $\kappa(G)=2$. If for each vertex cut set $\{u,v\}$ in $G$, there exists a component of $G-\{u,v\}$ such that its minimum degree is at most $g-1$, then $\kappa^{g}(G)\geq 3$, a contradiction. So there exists a vertex cut set $\{u,v\}$ in $G$ such that the minimum degree of each component of $G-\{u,v\}$ is at least $g$, as desired.

Suppose $\kappa(G)=1$. Then we have the following claim.

Claim 5**.**

$g\geq 1$.

Proof.

Assume, to the contrary, that $g=0$. Since $\kappa(G)=1$, it follows that there exists a cut vertex $v$ and the minimum degree of each component of $G-v$ is at least [math], and hence $\kappa^{0}(G)\leq 1$, which contradicts to the fact $\kappa^{g}(G)=2$. ∎

From Claim 5, we have $g\geq 1$. Since $\kappa^{g}(G)=2$, we have the following facts.

Fact 1**.**

For any cut vertex $v$, there exists a component of $G-v$ such that its minimum degree is at most $g-1$.

Fact 2**.**

There exist two vertices $x,y$ in $G$ such that $G-\{x,y\}$ is not connected and the minimum degree of each component of $G-\{x,y\}$ is at least $g$.

Suppose that one of $x,y$ is a cut vertex of $G$. Without loss of generality, we assume that $x$ is a cut vertex of $G$. Let $C_{1},C_{2},\ldots,C_{r}$ be the components of $G-x$.

Claim 6**.**

At most one of $C_{1},C_{2},\ldots,C_{r}$ has exactly one vertex.

Proof.

Assume, to the contrary, that there exist $C_{i},C_{j}\ (1\leq i\neq j\leq r)$ such that $|V(C_{i})|=|V(C_{j})|=1$. Then at least one component of $G-x-y$ is a isolated vertex, which contradicts to Fact 2. ∎

From Claim 6, if one of $C_{1},C_{2},\ldots,C_{r}$, say $C_{1}$, has exactly one vertex, then $C_{1}=\{y\}$ and $(2.2)(b)$ holds. Suppose that each $C_{i}\ (1\leq i\leq r)$ has at least two vertices.

Claim 7**.**

Exactly one of $C_{1},C_{2},\ldots,C_{r}$ has minimum degree at most $g-1$.

Proof.

Assume, to the contrary, that there exist $C_{i},C_{j}\ (1\leq i\neq j\leq r)$ such that $\delta(C_{i})\leq g-1$ and $\delta(C_{j})\leq g-1$. Then there is a component of $G-x-y$ such that its minimum degree is at most $g-1$, which contradicts to Fact $2$. ∎

From Claim 7, exactly one of $C_{1},C_{2},\ldots,C_{r}$, say $C_{1}$, has minimum degree at most $g-1$. Then there exists a vertex $u$ of degree at most $g-1$. We claim that $u=y$. Assume, to the contrary, that the degree of $u$ is most $g-1$ in $G-x-y$, a contradiction. Then $u=y$. From Fact 1, $(2.2)(a)$ holds.

Suppose that neither $x$ nor $y$ is a cut vertex of $G$. From Fact 1, $G-\{x,y\}$ is not connected and the minimum degree of each component is at least $g$. Then $(2.2)(c)$ holds. ∎

For $(1)$, $(2.2)(a)$, $(2.2)(b)$ and $(2.2)(c)$, we show the following examples corresponding them.

Example 4.1. Let $H_{1}$ be a graph obtained from $K_{g+1}$ and $K_{n-g-3}$ by adding two new vertices $u,v$ and edges in $E_{H_{1}}[u,V(K_{g+1})]\cup E_{H_{1}}[u,V(K_{n-g-3})]\cup E_{H_{1}}[v,V(K_{g+1})]\cup E_{H_{1}}[v,V(K_{n-g-3})]$, where $n\geq 2g+4$. From Theorem 3.2, $\kappa(H_{1})=\kappa_{g}(H_{1})=2$.

Example 4.2. Let $H_{2}$ be a graph obtained from $K_{g+1}$ and $K_{n-g-3}$ by adding two new vertices $u,v$ and edges in $\{uv,vv_{1},uu_{1}\}$, where $u_{1}\in V(K_{n-g-3})$ and $v_{1}\in V(K_{g+1})$ and $n\geq 2g+4$. From Theorem 3.2, $\kappa(H_{2})=1$ and $\kappa_{g}(H_{2})=2$.

Example 4.3. Let $H_{3}$ be a graph obtained from $K_{g+1}$ and $K_{n-g-3}$ by adding two new vertices $u,v$ and edges in $\{uv,vv_{1},vv_{2}\}$, where $v_{1}\in V(K_{n-g-3})$ and $v_{2}\in V(K_{g+1})$ and $n\geq 2g+4$. From Theorem 3.2, $\kappa(H_{3})=1$ and $\kappa_{g}(H_{3})=2$.

Example 4.4. Let $H_{4}$ be a graph obtained from $K_{g+1}$ and $K_{n-g-1}$ by adding edges in $\{vv_{1},vv_{2}\}$, where $v\in V(K_{g+1})$ and $v_{1},v_{2}\in V(K_{n-g-1})$ and $n\geq 2g+4$. From Theorem 3.2, $\kappa(H_{4})=1$ and $\kappa_{g}(H_{4})=2$.

4 Extremal problems

We now consider the three extremal problems that we stated in the Introduction.

Suppose that $n,k,g$ are three integers such that $(n-k)g$ is even and $2\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$. Let $F_{1},F_{2}$ be two $g$-regular graphs such that $|V(F_{1})|+|V(F_{2})|=n-k$. Let $F^{k}_{n}$ be a graph obtained from $F_{1},F_{2}$ and a star $K_{1,k-1}$ with center $v$ such that $|E_{G}[v,F_{1}]|=|E_{G}[v,F_{1}]|=1$.

Lemma 4.1**.**

Let $n,g,k$ be three integers with $2\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$. If $(n-k)g$ is even, then

[TABLE]

Proof.

Let $X=V(K_{1,k-1})$. Then $F^{k}_{n}-X$ is not connected and the minimum degree of each component of $G-X$ is at least $g$, and hence $\kappa^{g}(F^{k}_{n})\leq k$. Let $\kappa^{g}(F^{k}_{n})=t$. It suffices to show $t\geq k$. From the definition of $\kappa^{g}(F^{k}_{n})$, there exists $X\subseteq V(F^{k}_{n})$ with $|X|=t$ such that the minimum degree of each component of $F^{k}_{n}-X$ is at least $g$. Since $g\geq 2$, it follows that $V(K_{1,k-1})-v\subseteq X$. If $X=V(K_{1,k-1})-v$, then $G$ is connected, a contradiction. So $|X|\geq|V(K_{1,k-1})|-1+1=k$. So $\kappa^{g}(F^{k}_{n})\geq k$, and hence $\kappa^{g}(F^{k}_{n})=k$. ∎

Suppose that $n,k,g$ are three integers such that $(n-k)g$ is odd and $2\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$. Then $n-k$ is odd. Let $a,b$ be two integers such that $a$ is even, and $b$ is odd, and $a\geq g+1$, and $b\geq g+1$, and $a+b=n-k$. Let $H_{1}$ be a $g$-regular graph of order $a$. Let $H_{2}$ be a graph of order $b$ such that the degree of one vertex is exactly $g+1$, and the degree of each of the other vertices is exactly $g$. Let $H^{k}_{n}$ be a graph obtained from $H_{1},H_{2}$ and a star $K_{1,k-1}$ with center $v$ such that $|E_{G}[v,H_{1}]|=|E_{G}[v,G_{1}]|=1$. Similarly, we have the following lemma.

Lemma 4.2**.**

Let $n,g,k$ be three integers with $2\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$. Then

[TABLE]

Let $T_{n}^{\prime}$ be a tree of order $n$ obtained from three stars $K_{1,k-1},K_{1,a-1},K_{1,b-1}$ with centers $x,u,v$ by adding two edges $xu,xv$.

Lemma 4.3**.**

Let $n,k$ be two integers with $n\geq k+4$. Then

[TABLE]

Proof.

Let $X=V(K_{1,k-1})$. Then $T_{n}^{\prime}-X$ is not connected and the minimum degree of each component of $T_{n}^{\prime}-X$ is at least $1$, and hence $\kappa^{1}(T_{n}^{\prime})\leq k$. Let $\kappa^{1}(T_{n}^{\prime})=t$. It suffices to show $t\geq k$. From the definition of $\kappa^{1}(T_{n}^{\prime})$, there exists $X\subseteq V(T_{n}^{\prime})$ with $|X|=t$ such that the minimum degree of each component of $T_{n}^{\prime}-X$ is at least $1$. If $x\notin X$, then $u\in X$ or $v\in X$. Without loss of generality, let $u\in X$. Then $V(K_{1,a-1})\subseteq X$. Since $T_{n}^{\prime}-X$ is not connected and $x\notin X$, it follows that $v\in X$. Then $V(K_{1,b-1})\subseteq X$. Clearly, $T_{n}^{\prime}-X$ is connected, a contradiction. If $x\in X$, then $V(K_{1,k-1})\subseteq X$, and hence $|X|=t\geq k$. So $\kappa^{1}(T_{n}^{\prime})\geq k$, and hence $\kappa^{1}(T_{n}^{\prime})=k$. ∎

Theorem 4.1**.**

Let $n,g,k$ be three integers with $1\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$ and $1\leq k\leq n-2g-2$.

$(1)$* If $g=1$, then $s(n,k)=n-1$.*

$(2)$* If $n-k$ is even and $g\geq 2$, then*

[TABLE]

$(3)$* If $n-k$ is odd and $g\geq 2$, then*

[TABLE]

Proof.

$(1)$ Let $G=T_{n}^{\prime}$. From Lemma 4.3, we have $s(n,k)\leq n-1$. Since we only consider connected graphs, it follows that $s(n,k)\geq n-1$, and hence $s(n,k)=n-1$.

$(2)$ Suppose that $n-k$ is even. Let $G=F^{k}_{n}$. From Lemma 4.1, we have $s(n,k)\leq\frac{(n-k)g}{2}+k+1$. It suffice to show $s(n,k)\geq\frac{(n-k)g}{2}+k+1$. Let $G$ be a conneced graph of order $n$ with $\kappa_{g}(G)=k$ such that $e(G)$ is minimized. Then exists $X\subseteq V(G)$ with $|X|=k$ such that the minimum degree of each component of $G-X$ is at least $g$. Then $e(G-X)\geq\frac{(n-k)g}{2}$. Since $G$ is connected, it follows that $e(G)\geq\frac{(n-k)g}{2}+k+1$, and hence $s(n,k)=\frac{(n-k)g}{2}+k+1$.

$(3)$ Suppose that $n-k$ is odd. Let $G=H^{k}_{n}$. From Lemma 4.2, we have $s(n,k)\leq\frac{(n-k)g+1}{2}+k+1$. Similarly to the proof of $(1)$, we have $s(n,k)=\frac{(n-k)g+1}{2}+k+1$. ∎

Lemma 4.4**.**

Let $n,g,k$ be three integers with $1\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$. Let $G^{k}_{n}$ be the graph obtained from three cliques $K_{n-k-g},K_{k-1},K_{g+1}$ by adding the edges in $E_{G^{k}_{n}}[K_{n-k-g},K_{k-1}]\cup E_{G^{k}_{n}}[K_{g+1},K_{k-1}]$. Then

[TABLE]

Proof.

Let $X=K_{k-1}$. Since $1\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$, it follows that $G^{k}_{n}-X$ is not connected and each component has at least $g+1$ vertices, and hence $\kappa^{g}(G^{k}_{n})\leq k-1$. Clearly, $\kappa^{g}(G^{k}_{n})\geq\kappa(G^{k}_{n})=k-1$, and hence $\kappa^{g}(G^{k}_{n})=k-1$. ∎

Theorem 4.2**.**

Let $n,g,k$ be two integers with $1\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$ and $1\leq k\leq n-2g-2$. Then

[TABLE]

Proof.

To show $f(n,k)\geq{n\choose 2}-(n-k-g)(g+1)+1$, we construct $G^{k}_{n}$ defined in Lemma 4.4. Then $\kappa^{g}(G^{k}_{n})=k-1$. Since $|E(G^{k}_{n})|={n\choose 2}-(n-k-g)(g+1)$, it follows that $f(n,k)\geq{n\choose 2}-(n-k-g)(g+1)+1$.

Let $G$ be a graph with $n$ vertices such that $|E(G)|\geq{n\choose 2}-(n-k-g)(g+1)+1$. We claim that $\kappa^{g}(G)\geq k$. Assume, to the contrary, that $\kappa^{g}(G)\leq k-1$. Then there exists a vertex set $X\subseteq V(G)$ and $|X|\leq k-1$ such that the minimum degree of each component of $G-X$ is at least $g$. Let $C_{1},C_{2},\ldots,C_{r}$ be the components of $G-X$. The number of edges from $C_{1}$ to $C_{2}\cup C_{3}\cup\ldots\cup C_{r}$ in $\overline{G}$ is at least $|V(C_{1})|(n-|V(C_{1})|-|X|)\geq(n-k-g)(g+1)$ since $\sum_{i=1}^{r}|V(C_{i})|\geq n-k+1$ and $|V(C_{i})|\geq g+1$ for each $i\ (1\leq i\leq r)$. Clearly, $|E(G)|\leq{n\choose 2}-(n-k-g)(g+1)$, which contradicts to $|E(G)|\geq{n\choose 2}-(n-k-g)(g+1)+1$. So $\kappa^{g}(G)\geq k$, and hence $f(n,k)\leq{n\choose 2}-(n-k-g)(g+1)+1$.

From the above argument, we have $f(n,k)={n\choose 2}-(n-k-g)(g+1)+1$. ∎

Note that $g(n,k)=s(n,k+1)-1$. So we have the following.

Proposition 4.1**.**

Let $n,g,k$ be three integers with $2\leq g\leq\left\lfloor\frac{n-k-2}{2}\right\rfloor$ and $1\leq k\leq n-2g-2$.

$(1)$* If $n-k$ is odd and $g\geq 2$, then $g(n,k)=\frac{(n-k-1)g}{2}+k+1$.*

$(2)$* If $n-k$ is even and $g\geq 2$, then $g(n,k)=\frac{(n-k-1)g+1}{2}+k+1$.*

5 Concluding Remark

In this paper, we focus our attention on the $g$-good neighbor connectivity of general graphs. We have proved that $1\leq\kappa^{g}(G)\leq n-2g-2$ for $0\leq g\leq\min\{\Delta(G),\left\lfloor\frac{n-3}{2}\right\rfloor\}$. Trees with $\kappa^{g}(T_{n})=n-t$ are characterized in this paper. But the graphs with $\kappa^{g}(G)=n-t$ is still unknown. From Proposition 2.1, the classical $\kappa(G)$ is a natural lower bound of $\kappa^{g}(G)$, but there is no upper bound of $\kappa^{g}(G)$ in terms of $\kappa(G)$.