A note on sub-total domination in graphs

TL;DR

This paper introduces the sub-total domination number, a new graph invariant based on degree sequences, which provides a lower bound for the well-known total domination number in graphs without isolated vertices.

Contribution

The paper defines the sub-total domination number and establishes its role as a lower bound for the total domination number in simple graphs.

Findings

Sub-total domination number is a new degree sequence derived invariant.

It serves as a lower bound for the total domination number.

The concept applies to graphs without isolated vertices.

Abstract

Let $G$ be a simple and finite graph without isolated vertices. In this paper we introduce and study a new degree sequence derived invariant called the \emph{sub-total domination number}, denoted $\sub_t(G)$. In particular, we show that $\sub_t(G)$ serves as a lower bound on $\gamma_t(G)$, where $\gamma_t(G)$ denotes the heavily studied \emph{total domination number} of $G$.

Equations

$${\gamma }_t(G)\ge {\mathrm{sub}}_t(G),$$

$$n=\Big{|}\bigcup_{v\in S}N_{G}(v)\Big{|}\leq\sum_{v\in S}|N_{G}(v)|=\sum_{i=1}^{|S|}d_{G}(s_{i}).$$

$$\sum _{i=1}^{|S|}{d}_G(s_i)\ge n.$$

$$\sum _{i=1}^{|S|}{d}_i\ge \sum _{i=1}^{|S|}{d}_G(s_i)\ge n.$$

$$\sum _{i=1}^{|S|}{d}_i\ge n.$$

$$\sum _{i=1}^{{\mathrm{sub}}_t(G)}{d}_i\ge n.$$

$${\mathrm{sub}}_t(G)\Delta (G)=\sum _{i=1}^{{\mathrm{sub}}_t(G)}\Delta (G)\ge \sum _{i=1}^{{\mathrm{sub}}_t(G)}{d}_i\ge n,$$

$${\mathrm{sub}}_t(G)k=\sum _{i=1}^{{\mathrm{sub}}_t(G)}k\ge n.$$

$$\sum _{i=1}^{{\mathrm{sub}}_t(G)}{d}_i^G\ge n_1,$$

$$\sum _{i=1}^{{\mathrm{sub}}_t(H)}{d}_i^H\ge n_2.$$

$$\sum _{i=1}^{{\mathrm{sub}}_t(G)}{d}_i^G+\sum _{i=1}^{{\mathrm{sub}}_t(H)}{d}_i^H\ge n_1+n_2=n(G\cup H).$$

$$\sum _{i=1}^{{\mathrm{sub}}_t(G)+{\mathrm{sub}}_t(H)}{d}_i^{\ast }\ge \sum _{i=1}^{{\mathrm{sub}}_t(G)}{d}_i^G+\sum _{i=1}^{{\mathrm{sub}}_t(H)}{d}_i^H\ge n_1+n_2=n(G\cup H).$$

$$\sum _{i=1}^{{\mathrm{sub}}_t(G)+{\mathrm{sub}}_t(H)}{d}_i^{\ast }\ge n(G\cup H).$$

$${\gamma }_t(G\cup H)\ge {\mathrm{sub}}_t(G)+{\mathrm{sub}}_t(H)\ge {\mathrm{sub}}_t(G\cup H).$$

Full text

**Keywords: Total dominating sets; total domination number; sub-total domination number, degree sequence index strategy

AMS subject classification: 05C69**

1 Introduction

Domination in graphs is widely studied and a heavily applied notion in graph theory. Indeed, domination and its variants and generalizations appear in vast quantities in the mathematical literature; see for example [3, 6, 8, 12, 13, 14, 16, 24, 26]. Of the many variants of domination, total domination is arguably one of the most natural. Given a graph $G$, and a set of vertices $S$ in $G$, $S$ is a total dominating set if every vertex in $G$ has a neighbor in $S$. The minimum cardinality of a total dominating set in $G$ is the total domination number of $G$, denoted by $\gamma_{t}(G)$. It is well known that determining the total domination number of a general graph is in the class of $NP$-complete decision problems [25], and as such, a significant amount of research has been devoted to finding easily computable upper and lower bounds on $\gamma_{t}(G)$; see for example the monograph [16] which details and surveys total domination.

As previously mentioned, finding computationally efficient bounds on $\gamma_{t}(G)$ is desired. However, in a much more general fashion, it is of great interest to find computationally efficient bounds for any $NP$-hard graph invariant. With this in mind, we make note that the degree sequence of a graph has been shown to yield such desired bounds. Two well known examples are the residue and the annihilation number of a graph, which serve as respective lower and upper bounds on the computationally difficult independence number of a graph [7, 23]. With regards to domination, the lesser known degree sequence derived invariants known as the slater number and the sub-k-domination number serve as respective lower bounds on the domination number and k-domination number of a graph [1, 27]. We remark that these degree sequence results are special cases of the recently introduced degree sequence index strategy (DSI-strategy) [2].

Definitions and Notation. All graphs in this paper will be considered finite simple graphs without isolated vertices. Let $G=(V,E)$ be a graph. We will denote the order and size of $G$ by $n=n(G)=|V(G)|$ and $m=m(G)=|E(G)|$, respectively. When the dependence on $G$ is clear, we will write $n$ in place of $n(G)$. Two vertices $v,w\in V(G)$ are said to be neighbors if $vw\in E(G)$. The open neighborhood of $v\in V(G)$, denoted by $N_{G}(v)$, is the set of neighbors of $v$, whereas the close neighborhood of $v$ is the set $N_{G}[v]=N_{G}(v)\cup\{v\}$. The degree of $v\in V(G)$ is the cardinality of $N_{G}(v)$, and will be denoted by $d_{G}(v)$. The maximum and minimum vertex degree among all vertices of $G$ will be denoted by $\Delta(G)$ and $\delta(G)$, respectively. A graph $G$ is called $k$-regular if $d_{G}(v)=k$ for all $v\in V(G)$. A regular graph is a graph that is $k$-regular for some integer $k\geq 0$.

The degree sequence of $G$, is the sequence consisting of the vertex degrees in $G$ listed in non-increasing order, and will be denoted $D(G)=\{\Delta(G)=d_{1},\dots,d_{n}=\delta(G)\}$. For brevity, we may write the number of vertices realizing each degree in superscript. For example, the path $P_{n}$, on $n$ vertices, may have degree sequence written $D(P_{n})=\{2^{n-2},1^{2}\}$. If a sequence of non-negative integers $D$ has the property that $D=D(G)$, for some graph $G$, then we say that $D$ is a graphic sequence, and that $D$ is realizable by $G$. We note that a given graphic sequence may have more than one graph which realizes $D$.

A set of vertices $S\subseteq V(G)$ is a total dominating if every vertex in $G$ has a neighbor in $S$, and such a set will be called a TD-set of $G$. The cardinality of a smallest TD-set in $G$ is the total domination number of $G$, denoted by $\gamma_{t}(G)$, and such a set will be called a $\gamma_{t}(G)$-set. For other graph terminology and definitions, we will follow [16].

We will also make use of the notation $[k]=\{1,\dots,k\}$.

2 Sub-total domination

In this section we present our main results. First we recall the definition of the sub-total domination number, originally defined in [10], and denoted $sl_{t}(G)$. Keeping our notation and terminology consistent with [1], we will use $\mathrm{sub}_{t}(G)$ in place of $sl_{t}(G)$.

Definition 1

If $G$ is an isolate-free graph with order $n$ and degree sequence $D(G)=\{\Delta(G)=d_{1},\dots,d_{n}=\delta(G)\}$, the sub-total domination number $\mathrm{sub}_{t}(G)$, is defined as the smallest integer $k$ such that $\sum_{i=1}^{k}d_{i}\geq n$.

With the definition of sub-total domination now defined, we remark that $\mathrm{sub}_{t}(G)$ can be computed in $O(n)$ time. Because of the simplicity of computing $\mathrm{sub}_{t}(G)$, and the difficulty of computing $\gamma_{t}(G)$, the following theorem serves as one of our main results. We remark that this theorem first appeared in [10] without proof.

Theorem 1** ([10])**

If $G$ is an isolate-free graph, then

[TABLE]

and this bound is sharp.

**Proof. **Let $G$ be a graph with order $n$, degree sequence $D(G)=\{\Delta(G)=d_{1},\dots,d_{n}=\delta(G)\}$, and $S$ be a $\gamma_{t}(G)$-set. Next, we order the vertices of $S$, $s_{1},\dots,s_{|S|}$, so that $d_{G}(s_{1})\geq\dots\geq d_{G}(s_{|S|})$. By definition, every vertex is totally dominated by a vertex in $S$; that is, every vertex has a neighbor in $S$. Thus, $V(G)=\cup_{v\in S}N_{G}(v)$, which implies,

[TABLE]

In particular, we have established,

[TABLE]

Next observe that the $i$-th term of $D(G)$ is greater than or equal to the $i$-the degree of the list of vertices from $S$, and thus, we have the following inequality,

[TABLE]

That is,

[TABLE]

Since $\mathrm{sub}_{t}(G)$ is the smallest integer satisfying (1), it follows that $\gamma_{t}(G)=|S|\geq\mathrm{sub}_{t}(G)$, and the lower bound has been proven.

To see that this bound is sharp, consider the star $K_{1,n-1}$ on $n\geq 2$ vertices. Then, $\gamma_{t}(K_{1,n-1})=2$, and $\mathrm{sub}_{t}(K_{1,n-1})=2$. $\Box$

Theorem 1 is sharp for non-trivial stars. However, stars are a special case of a more general concept. Namely, if $G$ is a connected graph with order $n\geq 2$ and maximum degree $\Delta(G)=n-1$, then choosing a maximum degree vertex and an arbitrary neighbor of this vertex forms a TD-set, and hence, $\gamma_{t}(G)=2$. Moreover, the highest vertex degree summed with the next highest vertex degree will be greater than $n$, and so $\mathrm{sub}_{t}(G)=2$. In particular, since no vertex of $G$ will have degree $n$, it follows that $\mathrm{sub}_{t}(G)\geq 2$. We combine these ideas with the following proposition.

Proposition 2

If $G$ is a connected graph with order $n\geq 2$ and maximum degree $\Delta(G)=n-1$, then $\gamma_{t}(G)=\mathrm{sub}_{t}(G)=2$.

There exists graphs $G$ for which $\gamma_{t}(G)=2$ and $\Delta(G)\neq\Delta(G)-1$. Double stars (trees with exactly two non-leaf vertices) are one such example. With this in mind, we next generalize Proposition 2 to a statement on graphs $G$ with $\gamma_{t}(G)=2$. That is, since $\mathrm{sub}_{t}(G)\geq 2$, we obtain the following proposition.

Proposition 3

If $G$ is an isolate-free graph with $\gamma_{t}(G)=2$, then $\gamma_{t}(G)=\mathrm{sub}_{t}(G)$.

A simple lower bound on the total domination number of isolate-free graphs can be found by dividing the order by the maximum degree, see Chapter 2, Theorem 2.11. in [16]. With the following theorem we show that the sub-total domination number improves on this bound.

Theorem 4

If $G$ is an isolate-free graph with order $n$ and maximum degree $\Delta(G)$, then $\gamma_{t}(G)\geq\mathrm{sub}_{t}(G)\geq n/\Delta(G)$.

**Proof. **Let $G$ be an isolate-free graph with order $n$ and maximum degree $\Delta(G)$. The left hand side of the inequality is a restatement of Theorem 1. Thus, in order to prove this result, it suffices to show $\mathrm{sub}_{t}(G)\geq n/\Delta(G)$. By definition, we have

[TABLE]

Next observe that $\Delta(G)\geq d_{i}$ for each $i\in[\mathrm{sub}_{t}(G)]$, and thus

[TABLE]

Hence, $\mathrm{sub}_{t}(G)\geq n/\Delta(G)$, and the proof of the theorem is complete. $\Box$

3 Properties of $\mathrm{sub}_{t}(G)$

In this section we provide various fundamental properties of the sub-total domination number. We begin with a closed formula for $\mathrm{sub}_{t}(G)$ in the case that $G$ isolate-free and $k$-regular.

Proposition 5

If $k\geq 1$ is an integer and $G$ is a $k$-regular graph with order $n$, then $\mathrm{sub}_{t}(G)=\lceil n/k\rceil$.

**Proof. **Let $k\geq 1$ be an integer, and let $G$ be a $k$-regular isolate-free graph with order $n$. By definition of sub-total domination, we have

[TABLE]

It follows that $\mathrm{sub}_{t}(G)\geq n/k$. Since $\mathrm{sub}_{t}(G)$ is the smallest integer satisfying this inequality, we obtain $\mathrm{sub}_{t}(G)=\lceil n/k\rceil$, and the proof of the proposition is complete. $\Box$

Next we consider sub-total domination of disjoint isolate-free graphs. In particular, we show that sub-total domination is subadditive with respect to disjoint unions of graphs.

Lemma 6

If $G$ and $H$ are isolate-free graphs, then $\mathrm{sub}_{t}(G)+\mathrm{sub}_{t}(H)\geq\mathrm{sub}_{t}(G\cup H)$.

**Proof. **Let $G$ and $H$ be disjoint graphs with degree sequences $D(G)=\{\Delta(G)=d_{1}^{G},\dots,d_{n_{1}}^{G}=\delta(G)\}$ and $D(H)=\{\Delta(H)=d_{1}^{H},\dots,d_{n_{2}}^{H}=\delta(H)\}$. By definition of sub-total domination, we have

[TABLE]

and,

[TABLE]

Thus,

[TABLE]

Denote the degree sequence of $G\cup H$ by $D(G\cup H)=\{\Delta(G\cup H)=d_{1}^{*},\dots,d_{n_{1}+n_{2}}^{*}=\delta(G\cup H)\}$. Since degree sequences are listed in non-increasing order, it follows that

[TABLE]

That is,

[TABLE]

Since $\mathrm{sub}_{t}(G\cup H)$ is the smallest integer satisfying (2), it follows that $\mathrm{sub}_{t}(G\cup H)\leq\mathrm{sub}_{t}(G)+\mathrm{sub}_{t}(H)$, and the proof of the lemma is complete. $\Box$

It is easy to see that the total domination number is additive with respect to unions of disjoint graphs; that is, for disjoint isolate-free graphs $G$ and $H$, $\gamma_{t}(G\cup H)=\gamma_{t}(G)+\gamma_{t}(H)$. With this in mind, the following theorem serves as an improvement on Theorem 1 when considering the union of disjoint graphs.

Theorem 7

If $G$ and $H$ are isolate-free graphs, then

[TABLE]

**Proof. **Let $G$ and $H$ be isolate-free graphs. By Lemma 6 $\mathrm{sub}_{t}(G)+\mathrm{sub}_{t}(H)\geq\mathrm{sub}_{t}(G\cup H)$. Moreover, since total domination is additive with respect to disjoint unions, $\gamma_{t}(G\cup H)=\gamma_{t}(G)+\gamma_{t}(H)$. By Theorem 1, $\gamma_{t}(G)\geq\mathrm{sub}_{t}(G)$ and $\gamma_{t}(H)\geq\mathrm{sub}_{t}(H)$. Thus, $\gamma_{t}(G\cup H)\geq\mathrm{sub}_{t}(G)+\mathrm{sub}_{t}(H)$, and the theorem is proven. $\Box$

4 Conclusion and Open Problems

In this note we have studied fundamental properties of $\mathrm{sub}_{t}(G)$. However, we have not studied many classes of graphs for which $\gamma_{t}(G)=\mathrm{sub}_{t}(G)$. Since $\mathrm{sub}_{t}(G)$ is easily computable, we suggest the following problem.

Problem 1

Characterize all graphs $G$ for which $\gamma_{t}(G)=\mathrm{sub}_{t}(G)$.

Problem 1 is surely difficult, and leads to the question of asking if determining a graph $G$ satisfies $\gamma_{t}(G)=\mathrm{sub}_{t}(G)$ is $NP$-complete. The analogous question for sub-domination and domination is is known to be $NP$-complete [10], and so this provides evidence that this may indeed be the case.

There exists many lower bounds on the total domination number of a graph, and it remains to be shown how sub-total domination compares with most of these bounds. Thus, we further suggest the following problem.

Problem 2

Compare $\mathrm{sub}_{t}(G)$ with known lower bounds on $\gamma_{t}(G)$.