$\beta$-Packing Sets in Graphs

TL;DR

This paper introduces the concept of $eta$-packing sets in graphs, a new parameter that complements $eta$-domination, and analyzes its properties and values across various graph classes.

Contribution

It defines $eta$-packing sets, explores their properties, and determines their values for several classes of graphs, expanding the understanding of graph parameters.

Findings

Determined $eta$-pack($G$) for multiple graph classes.

Established properties of $eta$-packing sets.

Provided bounds and characterizations for $eta$-pack($G$).

Abstract

A set $S\subseteq V$ is $\alpha$-dominating if for all $v\in V-S$, $|N(v) \cap S | \geq \alpha |N(v)|.$ The $\alpha$-domination number of $G$ equals the minimum cardinality of an $\alpha$-dominating set $S$ in $G$. Since being introduced by Dunbar, et al. in 2000, $\alpha$-domination has been studied for various graphs and a variety of bounds have been developed. In this paper, we propose a new parameter derived by flipping the inequality in the definition of $\alpha$-domination. We say a set $S \subset V$ is a $\beta$-packing set of a graph $G$ if $S$ is a proper, maximal set having the property that for all vertices $v \in V-S$, $|N(v) \cap S| \leq \beta |N(v)|$ for some $0 < \beta \leq 1.$ The $\beta$-packing number of $G$ ($\beta$-pack($G$)) equals the maximum cardinality of a $\beta$-packing set in $G$. In this research, we determine $\beta$-pack($G$) for several classes of graphs, and we explore some properties of $\beta$-packing sets. Keywords: $\beta$-packing, $\alpha$-domination, graph theory, graph parameters

Equations

$$|N(v)\cap S|\le \beta |N(v)|$$

$$\frac{|N(v_i)\cap S|}{\mathrm{deg}(v_i)}=0\le \frac{1}{2}.$$

$$\frac{|N(v_i)\cap S|}{\mathrm{deg}(v_i)}=\frac{1}{2}\le \frac{1}{2}.$$

$$\frac{|N(v_i)\cap S|}{\mathrm{deg}(v_i)}=0\le \frac{1}{2}.$$

$$\frac{|N(v_i)\cap S|}{\mathrm{deg}(v_i)}=\frac{1}{2}\le \frac{1}{2}.$$

$$\frac{|N(v_i)\cap S|}{\mathrm{deg}(v_i)}=1>\frac{1}{2}.$$

$$\beta \text{-pack}(P_n)=\{\begin{array}{ll}0& 0<\beta <\frac{1}{2}\\ n-2& \frac{1}{2}\le \beta <1\\ n-1& \beta =1\end{array}$$

$$|N(v)\cap S^{\prime }|=|S^{\prime }|=\lfloor \beta \cdot n\rfloor .$$

$$\frac{|N(v)\cap (S\cup S^{\prime })|}{|N(v)|}=\frac{|N(v)\cap (S^{\prime })|}{|N(v)|}=\frac{\lfloor \beta \cdot n\rfloor }{n}\le \frac{\beta \cdot n}{n}=\beta .$$

$$\frac{|N(v^{\prime })\cap (S\cup S^{\prime })|}{|N(v^{\prime })|}\le \beta .$$

$$\frac{|N(v)\cap (S\cup S^{\prime }\cup U)|}{|N(v)|}\ge \frac{|N(v)\cap (S\cup \{u\})|}{|N(v)|}=\frac{\lfloor \beta \cdot m\rfloor +1}{m}.$$

$$\frac{|N(v)\cap (S\cup S^{\prime }\cup U)|}{|N(v)|}>\beta ,$$

$$\frac{1}{2}\beta \text{-pack}(K_{3,3,3,3})=5>\lfloor \beta \cdot 3\rfloor +\lfloor \beta \cdot 3\rfloor +\lfloor \beta \cdot 3\rfloor +\lfloor \beta \cdot 3\rfloor =4.$$

$$\frac{|N(v)\cap S|}{|N(v)|}\le {\beta }_1\le {\beta }_2.$$

$$\frac{|N(v)\cap S^{\prime }|}{|N(v)|}\le \beta$$

$$\beta <\frac{1}{\Delta (G)}\le \frac{1}{\mathrm{deg}(v)}\le \frac{|N(v)\cap S|}{\mathrm{deg}(v)},$$

$$\frac{|N(v)\cap S|}{|N(v)|}=\frac{|N(v)|}{|N(v)|}=1>\beta .$$

$$0,\frac{1}{d_1},\frac{2}{d_1},...,\frac{d_1-1}{d_1},1$$

$$\frac{1}{d_2},\frac{2}{d_2},...,\frac{d_2-1}{d_2}$$

$$⋮⋮$$

$$\frac{1}{d_t},\frac{2}{d_t},...,\frac{d_t-1}{d_t}.$$

$${\gamma }_{\alpha }(G)\le \beta \text{-pack}(G)?\text{ or }{\gamma }_{\alpha }(G)\ge \beta \text{-pack}(G)?$$

$${\gamma }_{\alpha }(K_{m,n})=\min \{\lceil \alpha m\rceil +\lceil \alpha n\rceil ,m\}.$$

$${\gamma }_{\frac{1}{2}}(K_{1,10})=1<\frac{1}{2}\text{-pack}(K_{1,10})=5.$$

Full text

Keywords: $\beta$-packing, $\alpha$-domination, graph theory, graph parameters

Benjamin M. Case was partially supported by the National Science Foundation under grants DMS-1403062 and DMS-1547399.

1. Introduction

Let $G=(V,E)$ be a graph with vertex set $V=\{v_{1},v_{2},...,v_{n}\}$ and order $n=|V|$. The open neighborhood of a vertex $v$ is the set $N(v):=\{u\>|\>uv\in E\}$ of vertices $u$ that are adjacent to $v$; the closed neighborhood of $v$, $N[v]:=N(v)\cup\{v\}.$

A set $S\subseteq V$ is $\alpha$-dominating if for all $v\in V-S$, $|N(v)\cap S|\geq\alpha|N(v)|.$ The $\alpha$-domination number of $G$ equals the minimum cardinality of an $\alpha$-dominating set $S$ in $G$. Since being introduced by Dunbar, Hoffman, Laskar, and Markus [4] in 2000, $\alpha$-domination has been studied for various graphs and a variety of bounds have been developed, see [1, 8, 5, 7, 2]. In this paper, we present a new parameter that is motivated by flipping the inequality in $\alpha$-domination, known as the $\beta$-packing set.

Definition 1.1**.**

For a graph $G=(V,E)$, a set $S\subset V$ is a $\beta$-packing set of a graph $G$ if $S$ is a proper, maximal set having the property (which we call the $\beta$-packing property) that for all vertices $v\in V-S$,

[TABLE]

for some $0<\beta\leq 1.$ The $\beta$-packing number of $G$, $\beta$-pack($G$), equals the maximum cardinality of a $\beta$-packing set in $G$.

For example, we say that a set $S\subset V$ is a 1/2-beta packing set if $v\in V-S$, $\frac{|N(v)\cap S|}{|N(v)|}\leq$ 1/2 and is maximal. The 1/2-beta packing number equals the maximum cardinality of a 1/2-beta packing set in $G$.

Example 1.2**.**

In Figure 1 we show all of the 1/2-beta packing sets of the shown graph (up to symmetry). The $\beta$-packings sets are shown as the black filled vertices. Note that in each graph, no subset of $V-S$ can be added to $S$ while preserving both the $\beta$-packing property and keeping the $\beta$-packing set a proper subset. The largest cardinality of these sets is 2, so $\frac{1}{2}\beta$-pack(G) = 2.

2. Examples and $\beta$-Packing Sets for Classes of Graphs

To begin we will consider some examples of different classes of graphs and try to determine some patterns about the $\beta$-packing number. We start by looking at the 1/2-beta packing sets for paths and then generalize these results to all paths and cycles. A $\frac{1}{2}\beta$-packing for $P_{6}$ is show in Figure 2.

Proposition 2.1**.**

Given a path $P_{n}$ of length n $\geq$ 2, $\frac{1}{2}\beta\text{-pack}(P_{n})=n-2$ and $V-S$ is connected.

Proof.

Consider a path of length $n$, $P_{n}=(V,E).$ If $V-S$ is not connected, $S$ is not maximal, see Proposition 3.2 where we show this in general. Suppose $S\subset V$ and $\{v_{i},v_{i+1}\}=V-S$ for some $i\in[1,n-1]$. As $S$ is proper, it suffices to show that the $\beta$-packing property is fulfilled and that S is maximal. To show the former, consider the following cases:

• •

If $\deg(v_{i})=1$, then $N(v_{i})\cap S=\emptyset$ and

[TABLE]

• •

If $\deg(v_{i})=2$, then $N(v_{i})\cap S=\{v_{i-1}\}$ and

[TABLE]

• •

If $\deg(v_{i+1})=1$ then $N(v_{i+1})\cap S=\emptyset$ and

[TABLE]

• •

If $\deg(v_{i+1})=2$ then $N(v_{i+1})\cap S=\{v_{i+2}\}$ and

[TABLE]

Thus the $\beta$-packing property holds in all cases. Now, we need to show that $S$ is maximal. WLOG, suppose $V-S=\{v_{i}\}$. We will again consider cases:

• •

If $\deg(v_{i})=1$ then $N(v_{i})\cap S=\{v_{i+1}\}$ and

[TABLE]

• •

If $\deg(v_{i})=2$ then $N(v_{i})\cap S=\{v_{i-1},v_{i+1}\}$

[TABLE]

∎

The following three results cover all the possible values of $\beta$ and show what the corresponding value of $\beta$-pack$(P_{n})$ is.

Proposition 2.2**.**

For $\frac{1}{2}\leq\beta<1$ and $n\geq 2$, $\beta\text{-pack}(P_{n})=n-2$ and $V-S$ is connected.

Proof.

This follows the same proof as the $\frac{1}{2}\beta$-packing set. ∎

Proposition 2.3**.**

For $0<\beta<\frac{1}{2}$, $\beta\text{-pack}(P_{n})=0$.

Proof.

For any $v_{i}\in V$, $\deg(v_{i})=$ 1 or 2. This implies $\frac{|N(v_{i})\cap S|}{\deg(v_{i})}$ is either 0, $\frac{1}{2}$ or 1. But $\frac{|N(v_{i})\cap S|}{\deg(v_{i})}\leq\beta<\frac{1}{2}$, which implies $S=\emptyset$. So, $\beta\text{-pack}(P_{n})=0$. ∎

Proposition 2.4**.**

For $\beta=1$, $\beta\text{-pack}(P_{n})=n-1$.

Proof.

Letting $\beta=1$ means that for any $v_{i}$, $\frac{|N(v_{i})\cap S|}{\deg(v_{i})}\leq\beta$. As $S$ must be a proper subset, we have to leave one node out of $S$. Thus,

$\beta\text{-pack}(P_{n})=n-1$. ∎

Corollary 2.5**.**

Given a cycle $C_{n}$ of size $n\geq 3$,

[TABLE]

and $V-S$ is connected.

Proof.

Note that any path can be made into a cycle by adding an edge. Thus, the proof for a cycle is identical to that of a path except that we need only to consider the cases of degree 2. ∎

Next we will consider complete bipartite graphs and determine their $\beta$-packing numbers. An example of a $\beta$-packing set is shown in Figure 3 for $K_{4,5}$.

Proposition 2.6**.**

Let $K_{m,n}=(V_{m},V_{n},E)$ be a complete bipartite graph. Then for $\beta<1$, all $\beta$-packing sets $S\cup S^{\prime}$ have the same size, where $S\subset V_{m}\subset V,$ $S^{\prime}\subset V_{n}\subset V$, with $|S|=\lfloor\beta\cdot m\rfloor$ and $|S^{\prime}|=\lfloor\beta\cdot n\rfloor$. Thus, $\beta\text{-}pack(K_{m,n})=\lfloor\beta\cdot m\rfloor+\lfloor\beta\cdot n\rfloor.$

Proof.

Let $S\subset V_{m}$ be any subset of size $\lfloor\beta\cdot m\rfloor$ and $S^{\prime}\subset V_{n}$ be any subset of size $\lfloor\beta\cdot n\rfloor$. Since $\beta<1$, $\lfloor\beta\cdot m\rfloor<m$ and $\lfloor\beta\cdot n\rfloor<n$. Thus, $S\cup S^{\prime}$ is proper. It suffices to show that the $\beta$-packing property is fulfilled and that $S\cup S^{\prime}$ is maximal. Let $v\in V_{m}-S$. Then, $\deg(v)=n$ implies

[TABLE]

Thus,

[TABLE]

Now let $v^{\prime}\in V_{n}-S^{\prime}$. Then, following the same process, we see that

[TABLE]

Finally, we must show that $S\cup S^{\prime}$ is maximal. Suppose for contradiction there was a proper subset $S\cup S^{\prime}\cup U\subset V$ for which the $\beta$-packing property held with $\emptyset\neq U\subset V-(S\cup S^{\prime})$. $U$ must contain at least one vertex $u$. WLOG, let $u$ be in the side $u\in V_{m}\cap U.$ Then for $v\in V_{n}-(S^{\prime}\cup U)$,

[TABLE]

Note that $\beta\cdot m<\lfloor\beta\cdot m\rfloor+1$ which implies $\beta<\frac{\lfloor\beta\cdot m\rfloor+1}{m}$. Thus

[TABLE]

so $S\cup S^{\prime}$ is maximal and $\beta\text{-pack}(K_{m,n})=\lfloor\beta\cdot m\rfloor+\lfloor\beta\cdot n\rfloor$.

∎

Proposition 2.7**.**

For $\beta=1$, $\beta\text{-pack}(K_{m,n})=m+n-1$.

Proof.

As the $\beta$-packing set must be proper, we let all the nodes be in the $\beta$-packing set and then remove one. As $|K_{m,n}|=m+n$, $\beta\text{-pack}(K_{m,n})<|K_{m,n}|$. Adding another node to this set would be all of $K_{m,n}$, so $S\cup S^{\prime}$ is both proper and maximal.

Let $v\notin S\cup S^{\prime}$. Then, $\frac{|N(v)\cap(S\cup S^{\prime})|}{|N(v)|}=1$, since every other node is in $S\cup S^{\prime}$. Thus, $\beta\text{-pack}(K_{m,n})=m+n-1$. ∎

If we try to generalize these results to complete multipartite graphs, Proposition 2.6 does not generalize in the natural way, but Proposition 2.7 does.

Example 2.8**.**

Consider the complete multipartite graph $K_{3,3,3,3}$ and let $\beta=1/2$. A $\beta$-packing set is given by taking 1 vertex in each of three partitions and 2 vertices out of the forth partition, for a total of 5 vertices in $S$. One can check this gives

[TABLE]

Corollary 2.9**.**

For $\beta=1$, $\beta\text{-pack}(K_{n_{1},n_{2},...,n_{m}})=n_{1}+\cdots+n_{m}-1$.

Proof.

The proof is similar to the bipartite case. ∎

3. General Properties of $\beta$-packing sets

In this section we present several general properties about $\beta$-packing sets and the $\beta$-packing number. Our first property shows how the $\beta$-packing numbers corresponding to different $\beta$’s are related.

Proposition 3.1**.**

Let $0<\beta_{1}\leq\beta_{2}\leq 1$. Then $\beta_{1}$-pack($G$) $\leq\beta_{2}$-pack($G$).

Proof.

Consider $\beta_{1}$-pack($G$), for any $\beta_{1}$-packing set $S$, $\forall v\in V-S$,

[TABLE]

So any such $S$ is contained in a $\beta_{2}$-packing set and one could add vertices until $S$ becomes maximal w.r.t $\beta_{2}$. ∎

It was already seen in Proposition 2.1 for paths that the complement a $\beta$-packing set is connected. This is in fact a general property that holds for all graphs.

Proposition 3.2**.**

For any $\beta$-packing set $S$, $V-S$ is connected.

Proof.

If $V-S$ is not connected, then $S$ is not maximal since one of the components of $V-S$ could be added to $S$ to form $S^{\prime}$ and for all other $v\in V-S^{\prime}$ we still have the property

[TABLE]

and $S^{\prime}$ would still be proper. ∎

Proposition 3.3**.**

Let $\Delta(G)$ be the max degree of a vertex of a connected graph. If $\beta<\frac{1}{\Delta(G)}$, then $\beta$-pack$(G)=0$.

Proof.

Suppose $S$ is a nonempty $\beta$-packing set. For any vertex $v\in V-S$, if a neighbor is in a $\beta$-packing set $S$, then

[TABLE]

a contradiction. Thus no vertex has a neighbor in $S$. Therefore $S=\emptyset$. ∎

The next three properties investigate the question of which values for $\beta$ in the interval $0<\beta\leq 1$ are interesting to consider.

Proposition 3.4**.**

If $\beta=1$, then $\beta$-pack$(G)=n-1$.

Proof.

A $\beta$-packing set must be proper, but we can just leave out any one vertex. ∎

Proposition 3.5**.**

Let $G$ be connected. If $\beta<1$, then $\beta$-pack$(G)<n-1$.

Proof.

Suppose $\{v\}=V-S$. Then

[TABLE]

∎

Let us consider the following question a bit more.

Question 1**.**

Given a graph $G$ how many ”interesting” $\beta$’s are there to consider? By interesting we mean that as $\beta$ increases from 0 to 1 it is only at these values where the value of $\beta$-pack$(G)$ could change.

Let $\delta(G)=d_{1},...,d_{t}=\Delta(G)$ be the distinct degrees of vertices in the graph. Then we claim the possible interesting $\beta$’s are a subset of the following ratios:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

4. Related Parameters

The initial motivation for defining $\beta$-packing sets was from $\alpha$-domination, and it is natural to ask what relationships the two parameters may have with each other. One might ask if $\beta=\alpha$ weather

[TABLE]

The answer is neither one in general. We have by Proposition 2.3 that $\beta=\frac{1}{3}$-pack$(P_{n})=0$. But from [4, Prop. 1] that $\gamma_{\alpha=\frac{1}{3}}(P_{n})=\lceil\frac{n}{3}\rceil$. So this is an example of $\frac{1}{3}\text{-pack}(G)<\gamma_{\frac{1}{3}}(G)$.

On the other hand, we have that by Proposition 2.6 that if when $\beta<1$ $\beta\text{-pack}(K_{m,n})=\lfloor\beta\cdot m\rfloor+\lfloor\beta\cdot n\rfloor$. In [4, Prop. 4] they have the result that for $1\leq m\leq n$

[TABLE]

Thus if we let for example $m=1$, $n=10$, $\beta=\alpha=1/2$ we get than

[TABLE]

We think it is an interesting open direction of study to consider if there are different relationship between $\alpha$-domination and $\beta$-packing would be interesting to consider.

5. Conclusion

In conclusion, we have introduced the new graph parameter, the $\beta$-packing number, and studied some of its properties and given formulas for it for certain classes of graphs. Our motivation for defining $\beta$-packing sets comes for $\alpha$-domination, but we leave it as an open direction to investigate what relationships these two parameters have with each other. Other interesting open directions would include determining the value of the $\beta$-packing number for other classes of graphs and determining the computational complexity of finding $\beta$-packing sets or the $\beta$-packing number. We hope that this introductory paper and promising future directions will promote further interest in considering $\beta$-packing.