A characterization of trees having a minimum vertex cover which is also a minimum total dominating set

TL;DR

This paper provides a constructive characterization of trees that possess a vertex cover which is also a minimum total dominating set, linking two important graph concepts.

Contribution

It introduces a new characterization of trees with a $( au_t- au)$-set, connecting minimum vertex covers and total domination in trees.

Findings

Identifies trees with a $( au_t- au)$-set.

Provides a constructive method for characterization.

Bridges concepts of vertex cover and total domination.

Abstract

A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(\gamma_t-\tau)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(\gamma_t-\tau)$-set.

Figures

Equations

$$\max \{{\gamma }_t(G),{\gamma }_t(H)\}\le {\gamma }_t(G{+}_{uv}H)\le {\gamma }_t(G)+{\gamma }_t(H),$$

$$\max \{\tau (G),\tau (H)\}\le \tau (G{+}_{uv}H)\le \tau (G)+\tau (H).$$

Full text

1 Introduction

Throughout this paper $G=(V,E)$ will be a finite, undirected, simple and connected graph of order $n$. The neighborhood of a vertex $v\in V$ is the set $N(v)$ of all vertices adjacent to $v$ in $G$. For a set $X\subseteq V,$ the open neighborhood, $N(X)$, is defined to be $\bigcup_{v\in X}N(v)$ and the closed neighborhood of $X$ is defined as $N[X]=N(X)\cup X.$ The degree of a vertex $v\in V$ is $d(v)=|N(v)|$. A vertex $v\in V$ is an end vertex if $d(v)=1.$ A support vertex, or support, is the neighbor of an end vertex; a strong support vertex is the neighbor of at least two end vertices. For a set $S\subseteq V,$ and $v\in S$, the private neighborhood $\textnormal{pn}(v,S)$ of $v\in S$ is defined by $\textnormal{pn}(v,S)=\{u\in V\colon\ N(u)\cap S=\{v\}\}$. Each vertex in $\textnormal{pn}(v,S)$ is called a private neighbor of $v$.

A vertex cover of $G$ is a set $X\subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A minimum vertex cover is a vertex cover of smallest possible cardinality. The vertex cover number of $G$, $\tau(G)$, is the cardinality of a minimum vertex cover of $G$. A vertex cover of cardinality $\tau(G)$ is called a $\tau(G)$-set.

The minimum vertex cover problem arises in various important applications, including multiple sequence alignments in computational biochemistry (see for example [15]). In computational biochemistry there are many situations where conflicts between sequences in a sample can be resolved by excluding some of the sequences. Of course, exactly what constitutes a conflict must be precisely defined in the biochemical context. It is possible to define a conflict graph where the vertices represent the sequences in the sample and there is an edge between two vertices if and only if there is a conflict between the corresponding sequences. The aim is to remove the fewest possible sequences that will eliminate all conflicts, which is equivalent to finding a minimum vertex cover in the conflict graph $G.$ Several approaches, such as the use of a parameterized algorithm [4] and the use of a simulated annealing algorithm [17], have been developed to deal with this problem.

A subset $D$ of $V$ is dominating in $G$ if $N[D]=V$. The domination number of $G$, denoted by $\gamma(G)$, is the minimum cardinality among all dominating sets in $G.$ A dominating set $D$ is a total dominating set of $G$ if the subgraph $G[D]$ induced by $D$ has no isolates. In [2], Cockayne et al. defined the total domination number $\gamma_{t}(G)$ of a graph $G$ to be the minimum cardinality among all total dominating sets of $G$. A total dominating set of cardinality $\gamma_{t}(G)$ is called a $\gamma_{t}(G)$-set.

A total vertex cover is a set which is both a total dominating set and vertex cover. In [5], Dutton studies total vertex covers of minimum size. He proved that, in general, the associated decision problem is $\mathcal{NP}$-complete, and gives some bounds of the size of a minimum total vertex cover of a graph $G$ in terms of $\gamma_{t}(G)$ and $\tau(G)$; this parameter has received some attention in recent years [6, 13]. In this work, we explore a particular case of total vertex covers. A $(\gamma_{t}-\tau)$-set of $G$ is a total vertex cover which is both a $\gamma_{t}(G)$-set and a $\tau(G)$-set. While every graph has a total vertex cover, by considering $K_{2}$, it is trivial to observe that not every graph has a $(\gamma_{t}-\tau)$-set. So, it is natural to ask for a characterization of graphs having a $(\gamma_{t}-\tau)$-set.

Clearly, a graph $G$ having a $(\gamma_{t}-\tau)$-set also satisfies $\gamma_{t}(G)=\tau(G)$; a graph satisfying this equation will be called a $(\gamma_{t}-\tau)$-graph. Again, $K_{2}$ is an example of a graph which is not a $(\gamma_{t}-\tau)$-graph, and so, the following question arises: Does every $(\gamma_{t}-\tau)$-graph contains a $(\gamma_{t}-\tau)$-set? Unfortuately, the answer is no (consider the path on $8$ vertices, $P_{8}$). So, another natural problem to consider is to find a characterization of $(\gamma_{t}-\tau)$-graphs.

Total domination in graphs is well described in [9] and recently in [11] and [12]. Among the different variants of domination, total domination is probably the best known and the most widely studied. Total domination has been successfully related to many graph theoretic parameters [12]; in particular, an additional motivation for this work is the following observation. It is known that for every graph $G$, $\gamma(G)\leq\alpha^{\prime}(G)$, where $\alpha^{\prime}(G)$ is the matching number of $G$. Nonetheless, neither $\alpha^{\prime}(G)$ nor $\gamma_{t}(G)$ bounds the other one, and it is an interesting problem to find families of graphs $G$ such that $\gamma_{t}(G)\leq\alpha^{\prime}(G)$, [12]. On the other hand, in [7], Hartnell and Rall characterized all the graphs $G$ such that $\gamma(G)=\tau(G)$. Recalling that for every bipartite graph $G$ we have $\tau(G)=\alpha^{\prime}(G)$, it is natural to consider the problem of characterizing bipartite graphs $G$ such that $\gamma_{t}(G)=\tau(G)$. Since trees are the best-known bipartite graphs, the problem of characterizing the trees $T$ such that $\gamma_{t}(G)=\tau(G)$ seems to be a very good one.

A usual approach in the literature for characterizing families of trees with a certain property is to consider a constructive characterization. First, a family $B$ of trees having the property $P$ (where it is usually trivial to verify it) is chosen as a (recursive) base, and then, some operations preserving $P$ are introduced. Finally, it is proved that the family of trees having the property $P$ are precisely those trees that can be constructed from a tree in $B$ by recursive applications of the proposed operations. This approach has been used extensively, to characterize, for example, Roman trees [10], trees with equal independent domination and restrained domination numbers, trees with equal independent domination and weak domination numbers [8], trees with equal independent domination and secure domination numbers [14], trees with at least $k$ disjoint maximum matchings [16], trees with equal $2$-domination and $2$-independence numbers [1], trees with equal domination and independent domination numbers, trees with equal domination and total domination numbers [3], etc. In [3], a general framework for studying constructive characterizations of trees having an equality between two parameters is discussed.

The main goal of this article is to provide a constructive characterization of the trees having a $(\gamma_{t}-\tau)$-set. For unexplained terms and symbols we refer the reader to [9]. The rest of the paper is structured as follows. In Section 2 we present some basic results that will be used in the rest of the paper; it is also proved that the difference between $\gamma_{t}(G)$ and $\tau(G)$ can be arbitrarily large. Section 3 is devoted to prove our main result, we show that the family of trees $T$ having a $(\gamma_{t}-\tau)$-set can be constructed through four simple operations starting from $P_{4}$. In the final section some related problems are proposed.

2 Basic results relating $\gamma_{t}(G)$ and

$\tau(G)$

In Section 3 we will define four operations which will be used to construct all the trees having a $(\gamma_{t}-\tau)$-set. Such operations will be defined using the following definition.

Definition 1**.**

Let $G=(V(G),E(G))$ and $H=(V(H),E(H))$ be two disjoint graphs, and let $u$ and $v$ be vertices in $V(G)$ and $V(H)$, respectively. The sum of $G$ with $H$ via the edge $uv$, $G+_{uv}H$, is defined as $V(G+_{uv}H)=V(G)\cup V(H)$ and $E(G+_{uv}H)=E(G)\cup G(H)\cup\{uv\}$.

Moreover, if $H=K_{1}=\{v\}$, we say that we add $v$ to $G$ supported by $u$.

Let $G$ and $H$ be two graphs with $u\in G$ and $v\in H$. Notice that, regardless of the choice of $u$ and $v$, the following inequalities are always satisfied:

[TABLE]

[TABLE]

It is also worth noticing that, for each of the previous four inequalities, there are examples where they are strict, and examples where they are equalities; we will come across them in the following sections.

We will now use the previously defined sum to prove that the difference between $\gamma_{t}$ and $\tau$ can be arbitrarily large, even for trees.

Proposition 1**.**

For any positive integer $k$ there exists a tree $T_{(k)}$ such that $\tau(T_{(k)})-\gamma_{t}(T_{(k)})=k.$

Proof.

Let $P_{4k+2}=(v_{1},v_{2},\ldots,v_{4k+2})$ be a path. Add $2k+2$ new vertices to $P_{4k+2}$ supported by the $2k+2$ vertices $\{v_{1},v_{2},v_{5},v_{6},v_{9},v_{10},\ldots,v_{4k+1},v_{4k+2}\}$. The graph that we obtain is a tree $T_{(k)}$ such that $\gamma_{t}(T_{(k)})=2k+2$, and $\tau(T_{(k)})=3k+2$. Thus, we have $\tau(T_{(k)})-\gamma_{t}(T_{(k)})=k.$ See Figure 1. ∎

Proposition 2**.**

For every positive integer $k$ there exists a tree $T^{\prime}_{(k)}$ such that $\gamma_{t}(T^{\prime}_{(k)})-\tau(T^{\prime}_{(k)})=k.$

Proof.

Let $P_{4k-1}=(v_{1},v_{2},\ldots,v_{4k-1})$ be a path. Add $2k$ new vertices to $P_{4k-1}$ supported by the vertices with odd index. The graph that we obtain is a tree $T^{\prime}_{(k)}$ such that $\gamma_{t}(T^{\prime}_{(k)})=3k,\tau(T^{\prime}_{(k)})=2k$. Hence, $\gamma_{t}(T^{\prime}_{(k)})-\tau(T^{\prime}_{(k)})=k.$ See Figure 2.

∎

The following simple remark will be useful in the proof of our main result.

Remark 3**.**

Let $G$ be a graph with at least three vertices. If $G$ is not a star, then there exists a minimum total dominating set $D\subseteq V(G)$ such that $D$ contains no end vertex of $G$.

Proof.

Let $D$ be a $\gamma_{t}(G)$-set and $x$ an end vertex of $G$ such that $N(x)=\{y\}$. Then $D-\{x\}\cup\{z\}$ is a total dominating set of $G,$ where $z\in N(y)$ is not an end vertex of $G$. ∎

Our next result will also be very useful in the following section.

Lemma 4**.**

If $\gamma_{t}(G)=\tau(G)$ and $D$ is a $(\gamma_{t}-\tau)$-set of $G$, then $D$ contains no end vertex of $G.$

Proof.

Let $D\subseteq V(G)$ be a ($\gamma_{t}-\tau$)-set of $G$. If $D$ contains an end vertex $x$, then, since $D$ is a total dominating set, it follows that there exists a vertex $y\in D\cap N_{G}(x)$. This implies that $D\setminus\{x\}$ is a vertex cover of $G$, a contradiction to the assumption that $\gamma_{t}(G)=\tau(G)$. ∎

As we mentioned in the introduction, not every tree contains a $(\gamma_{t}-\tau)$-set. The smallest tree having a $(\gamma_{t}-\tau)$-set is $P_{4}$, which also happens to be the smallest $(\gamma_{t}-\tau)$-tree. But not every $(\gamma_{t}-\tau)$-tree contains a $(\gamma_{t}-\tau)$-set. Actually, it is not hard to find an infinite class of $(\gamma_{t}-\tau)$-trees not having a $(\gamma_{t}-\tau)$-set, the most simple one is the family of paths $P_{4k}$, for $k\geq 2$. Thus, the class of tress having a $(\gamma_{t}-\tau)$-set is properly contained in the class of $(\gamma_{t}-\tau)$-trees.

Given a class of graphs, it is common in graph theory to aim for a characterization in terms of a set of forbidden induced subgraphs, because such characterization directly implies polynomial time recognition for the class. Unfortunately, neither $(\gamma_{t}-\tau)$-trees, nor trees having a $(\gamma_{t}-\tau)$-set, admit a characterization of this kind. To prove this fact, consider the following construction.

Recall that the corona of a graph $G$ is the graph obtained from $G$ by adding a new vertex $v^{\prime}$ to $G$ supported by $v$, for every vertex $v\in V(G)$. If $H$ is the corona of the graph $G$, then clearly $V(G)$ is a $(\gamma_{t}-\tau)$-set of $H$. Hence, any graph $G$ is an induced subgraph of a $(\gamma_{t}-\tau)$-graph (of a graph having a $(\gamma_{t}-\tau)$-set), and thus, there exists no forbidden subgraph characterization of $(\gamma_{t}-\tau)$-graphs (of graphs having a $(\gamma_{t}-\tau)$-set).

In our next section, we will obtain a constructive characterization of trees having a $(\gamma_{t}-\tau)$-set. For this end, we finish this section introducing a definition and proving a simple technical result.

Definition 2**.**

Let $G$ be a graph and $S$ a $\gamma_{t}(G)$-set. A vertex $v$ is $S$-quasi-isolated if there exists $u\in S$ such that $\textnormal{pn}(u,S)=\{v\}$. A vertex $v$ is quasi-isolated if it is $S$-quasi-isolated for some $\gamma_{t}(G)$-set $S$.

A vertex $v$ is a $2$-support if it is at distance two from an end vertex. The next proposition shows that if a vertex is a $2$-support, then it is not quasi-isolated.

Proposition 5**.**

Let $G$ be a graph and $v\in V$ a $2$-support. Then the vertex $v$ is non-quasi-isolated.

Proof.

Let $x,y,v\in V$ be a leaf, a support and a $2$-support of $G$, respectively, such that $y\in N(x)\cap N(v)$. For every $\gamma_{t}(G)$-set $S$, $y\in S$, $v\in N(y)$ and $x\in\textnormal{pn}(y,S)$, therefore for any $u\in S$, $\textnormal{pn}(u,S)\neq\{v\}$. Hence, $v$ is not quasi-isolated. ∎

3 Trees having a $(\gamma_{t}-\tau)$-set

As discussed in the previous section, trees having a $(\gamma_{t}-\tau)$-set do not admit a characterization through a set forbidden subgraphs. Following the usual approach in this kind of situation, we will propose a set of operations preserving the existence of a $(\gamma_{t}-\tau)$-set to obtain an infinite family of trees having a $(\gamma_{t}-\tau)$-set, and then, we will prove that every tree having a $(\gamma_{t}-\tau)$-set belongs to this family.

We define the family $\mathcal{T}$ of trees to consist of all trees $T$ that can be obtained from a sequence $T_{1},T_{2},\dots,T_{k}$ of trees such that $T_{1}$ is the path $P_{4}$, $T=T_{k}$ and, if $k\geq 2$, $T_{i+1}$ can be obtained recursively from $T_{i}$ by one of the following operations.

• •

Operation $\mathcal{O}_{1}$: Consider $u\in V(T)$ such that $u$ belongs to some $(\gamma_{t}-\tau)$-set. Let $v$ be a leaf of a path $P_{4}$. Then do the sum of $T$ with $P_{4}$ via the edge $uv$.

• •

Operation $\mathcal{O}_{2}$: Let $u\in V(T)$ such that $u$ belongs to some $(\gamma_{t}-\tau)$-set. Then add a new vertex $v$ to $T$ supported by $u$.

• •

Operation $\mathcal{O}_{3}$: Let $u\in V(T)$ such that $u$ belongs to some $(\gamma_{t}-\tau)$-set and $u$ it is not a quasi-isolated vertex. Let $P_{2}=(v,w)$ be a path with two vertices. Then do the sum of $T$ with $P_{2}$ via the edge $uv$.

• •

Operation $\mathcal{O}_{4}$: Let $u\in V(T)$ such that $u$ is not a quasi-isolated vertex of $T$. Let $v$ be a support vertex of a path $P_{4}$. Then do the sum of $T$ with $P_{4}$ via the edge $uv$.

Our next lemma is valid for any tree, not necessarily a tree in $\mathcal{T}$.

Lemma 6**.**

Let $T$ be a tree. If $T_{i}$ is a tree obtained from $T$ by an operation $\mathcal{O}_{i}$, $1\leq i\leq 4$, then:

$\gamma_{t}(T_{1})=\gamma_{t}(T)+2$* and $\tau(T_{1})=\tau(T)+2$;* 2. 2.

$\gamma_{t}(T_{2})=\gamma_{t}(T)$* and $\tau(T_{2})=\tau(T)$;* 3. 3.

$\gamma_{t}(T_{3})=\gamma_{t}(T)+1$* and $\tau(T_{3})=\tau(T)+1$;* 4. 4.

$\gamma_{t}(T_{4})=\gamma_{t}(T)+2$* and $\tau(T_{4})=\tau(T)+2$;*

and hence, $\gamma_{t}(T)-\tau(T)=\gamma_{t}(T_{i})-\tau(T_{i})$, for $1\leq i\leq 4$. In particular $\gamma_{t}(T)=\tau(T)$ if and only if $\gamma_{t}(T_{i})=\tau(T_{i})$, for $1\leq i\leq 4$.

Proof.

Observe that for $1\leq i\leq 4$, $\gamma_{t}(T_{i})\geq\gamma_{t}(T)$ and $\tau(T_{i})\geq\tau(T)$. We consider four cases.

• •

Suppose $i=1$, $P_{4}=(v,x,y,z)$ and $T_{1}=T+_{uv}P_{4}.$ Let $S$ be a $\gamma_{t}(T)$-set (a $\tau(T)$-set, respectively). Then, $S^{\prime}=S\cup\{x,y\}$ ($S^{\prime}=S\cup\{v,y\}$, resp.), is a total dominating set (vertex cover, resp.) of $T_{1}$. Thus, $\gamma_{t}(T_{1})\leq\gamma_{t}(T)+2$ and $\tau(T_{1})\leq\tau(T)+2.$

For purposes of contradiction, let $D$ be a $\gamma_{t}(T_{1})$-set such that $|D|\leq\gamma_{t}(T)+1.$ Define $S=D\cap V(P_{4})$, then $2\leq|S|\leq 3.$ Suppose $|S|=2$, then $v\notin D$ and $D-S$ is a total dominating set of $T$ with cardinality less than or equal to $\gamma_{t}(T)-1.$ If $|S|=3$, then $(D-S)\cup\{w\}$ for $w\in N_{T}(u)$ is a total dominating set of $T$ with cardinality less than or equal to $\gamma_{t}(T)-1.$ Therefore, $\gamma_{t}(T_{1})=\gamma_{t}(T)+2.$

For purposes of contradiction, let $D$ be a $\tau(T_{1})$-set such that $|D|\leq\tau(T)+1.$ Define $S=D\cap V(P_{4})$, then $|S|=2$. Suppose $S=\{x,y\},$ or $S=\{x,z\}$ or $S=\{v,y\}$, then $D-S$ is a vertex cover of $T$ with cardinality less than or equal to $\tau(T)-1.$ Hence, $\tau(T_{1})=\tau(T)+2.$

• •

For $i=2$ the proof is straightforward.

• •

Suppose $i=3$, $P_{2}=(v,w)$ and $T_{3}=T+_{uv}P_{2}.$ Let $S$ be a $\gamma_{t}(T)$-set such that $u\in S$, then $S^{\prime}=S\cup\{v\}$ is a total dominating set of $T_{3}$. Similarly, if $S$ is a $\tau(T)$-set then $S^{\prime}=S\cup\{v\}$ is a vertex cover of $T_{3}$. Thus, $\gamma_{t}(T_{3})\leq\gamma_{t}(T)+1$ and $\tau(T_{3})\leq\tau(T)+1.$

For purposes of contradiction, let $D$ be a $\gamma_{t}(T_{3})$-set such that $|D|=\gamma_{t}(T)$ and there is not end vertex in $D$ (such set exists by Remark 3). Then $D\cap V(P_{2})=\{v\}$ and $u\in D$. Since $|D-\{v\}|<\gamma_{t}(T)$, the set $D-\{v\}$ is not a total dominating set of $T$. But, for all $z\in N_{T}(u)$, the set $D^{\prime}=D-\{v\}\cup\{z\}$ is a $\gamma_{t}(T)$-set such that $u$ is $D^{\prime}$-quasi-isolated, a contradiction. So, $\gamma_{t}(T_{3})=\gamma_{t}(T)+1$.

By definition of vertex cover, it is not posible that $\tau(T_{3})=\tau(T)$, so $\tau(T_{3})=\tau(T)+1.$

• •

Suppose $i=4$, $P_{4}=(x,v,y,z)$ and $T_{4}=T+_{uv}P_{4}.$ Let $S$ be a $\gamma_{t}(T)$-set (a $\tau(T)$-set, respectively). Then, $S^{\prime}=S\cup\{v,y\}$ is a total dominating set (vertex cover, resp.) of $T_{4}$. Thus, $\gamma_{t}(T_{4})\leq\gamma_{t}(T)+2$ and $\tau(T_{4})\leq\tau(T)+2.$

For purposes of contradiction, let $D$ be a $\gamma_{t}(T_{4})$-set such that $|D|\leq\gamma_{t}(T)+1.$ Then $D\cap V(P_{4})=\{v,y\}$. Since $|D-\{v,y\}|\leq\gamma_{t}(T)-1$, the set $D-\{v,y\}$ is not a total dominating set of $T.$ But, for all $w\in N_{T}(u)$, the set $D^{\prime}=D-\{v,y\}\cup\{w\}$ is a $\gamma_{t}(T)$-set such that $u$ is $D^{\prime}$-quasi-isolated, a contradiction. So, $\gamma_{t}(T_{4})=\gamma_{t}(T)+2$.

By definition of vertex cover, it is not posible that $\tau(T_{4})\leq\tau(T)+1$, so $\tau(T_{3})=\tau(T)+2.$

∎

Corollary 7**.**

Suppose $T$ is a tree with $D$ a ($\gamma_{t}-\tau$)-set of $T$. If $T_{i}$ is a tree obtained from $T$ by an operation $\mathcal{O}_{i}$, $1\leq i\leq 4$, then $T_{i}$ has a ($\gamma_{t}-\tau$)-set $D_{i}$.

Proof.

Let $D$ be a $(\gamma_{t}-\tau)$-set of $T$. With the notation of the above lemma, we have:

• •

If $i=1$ then $D_{1}=D\cup\{x,y\}$.

• •

If $i=2$ then $D_{2}=D$.

• •

If $i=3$ then $D_{3}=D\cup\{v\}$.

• •

If $i=4$ then $D_{4}=D\cup\{v,y\}$.

∎

Theorem 8**.**

If $T\in\mathcal{T}$, then $T$ is a ($\gamma_{t}-\tau$)-tree.

Proof.

Let $T=P_{4}$, then $\gamma_{t}(T)=\tau(T)=2$. By Lemma 6 and Corollary 7, the proof is straightforward. ∎

Lemma 9**.**

Let $T$ be a tree and $u$ a vertex in $T$.

Let $P_{2}=(v,w)$ be a path of length two. Suppose that $u$ belongs to some $\gamma_{t}(T)$-set $D$ of $T$ and define $T^{\prime}$ to be the sum of $T$ with $P_{2}$ via the edge $uv$. If $u$ is $D$-quasi-isolated, then $\gamma_{t}(T)=\gamma_{t}(T^{\prime})$. 2. 2.

Let $v$ and $w$ be the support vertices of a path $P_{4}$. Define $T^{\prime}$ to be the sum of $T$ with $P_{4}$ via the edge $uv$. If $u$ is a quasi-isolated vertex, then $\gamma_{t}(T)=\gamma_{t}(T^{\prime})+1$.

Proof.

Let $D$ be a $\gamma_{t}(T)$-set such that $u$ is $D$-quasi-isolated. There exists $z\in D$ such that $\textnormal{pn}(z,D)=\{u\}$. It is easy to verify that $D^{\prime}=(D\setminus\{z\})\cup\{v\}$, is a $\gamma_{t}(T^{\prime})$-set, in the first case, and $D^{\prime}=(D\setminus\{z\})\cup\{v,w\}$ is a $\gamma_{t}(T^{\prime})$-set for the second case. ∎

Our main result is the following.

Theorem 10**.**

Let $T$ be a tree. If $T$ has a $(\gamma_{t}-\tau)$-set, then $T\in\mathcal{T}$.

Proof.

By induction on $n=|V(T)|$. Since $\gamma_{t}(T)=\tau(T),$ we have $n\geq 4.$ The only tree $T$ with four vertices and equality $\gamma_{t}(T)=\tau(T)$ is $P_{4}$, and $P_{4}\in\mathcal{T}$.

Let $T$ be a tree with $n>4$ and let $D$ be a ($\gamma_{t}-\tau$)-set of $T$. If $T$ has a strong support vertex $v$ with a leaf $u$, then $D$ is a ($\gamma_{t}-\tau$)-set of $T^{\prime}=T-\{u\}$. By induction hypothesis $T^{\prime}\in\mathcal{T}$ and, using operation $\mathcal{O}_{2}$ we have that $T\in\mathcal{T}$. Therefore we can assume that there are no strong support vertices in $T$.

Let $P=(v_{0},\ldots,v_{l})$ be a longest path in $T$. Then $d_{T}(v_{1})=2$ and by Lemma 4 the vertices $v_{1},v_{2}\in D$. The proof of the theorem follows to the next two claims.

Claim 1. If there exists a vertex $x\in N_{T}(v_{2})\cap D$ such that $x\neq v_{1}$ then $d_{T}(v_{2})\neq 2$ and $T\in\mathcal{T}$.

Proof of Claim 1. Observe that $d_{T}(v_{2})>2$. Otherwise, $d_{T}(v_{2})=2$ and hence $x=v_{3}$ and $D-\{v_{2}\}$ is a vertex cover of $T$, contradicting $\gamma_{t}(T)=\tau(T)$. If $T^{\prime}=T-\{v_{0},v_{1}\}$, then it is not hard to see that $D\setminus\{v_{1}\}$ is a $(\gamma_{t}-\tau)$-set of $T^{\prime}$. From the induction hypothesis $T^{\prime}\in\mathcal{T}$. For sake of contradiction, suppose that $v_{2}$ is quasi-isolated in $T^{\prime}$. By Lemma 9, $\gamma_{t}(T)=\gamma_{t}(T^{\prime})=\gamma_{t}(T)-1$, a contradiction. Therefore, $v_{2}$ is not quasi-isolated in $T^{\prime}$, and using operation $\mathcal{O}_{3}$, we have that $T\in\mathcal{T}$.

Claim 2. If $N_{T}(v_{2})\cap D=\{v_{1}\}$ then $d_{T}(v_{3})=2,d_{T}(v_{4})>2$ and $T\in\mathcal{T}$.

Proof of Claim 2. If $N_{T}(v_{2})\cap D=\{v_{1}\}$, then $v_{2}$ is a support vertex or $d_{T}(v_{2})=2.$

Observe that if $d_{T}(v_{3})=1$, since $T$ does not have strong support vertices, then $T=P_{4}.$ Therefore, $d_{T}(v_{3})\geq 2$. Since $D$ is a vertex cover of $T$ and $v_{3}\notin D$, $|N_{T}(v_{3})\cap D|\geq 2$.

Suppose $v_{2}$ is a support vertex and let $T^{\prime}$ be the tree $T^{\prime}=T-\{v_{0},v_{1},v_{2},x\},$ where $x$ is the leaf neighbour of $v_{2}.$ The set $D^{\prime}=D-\{v_{1},v_{2}\}$ is a ($\gamma_{t}-\tau$)-set of $T^{\prime}$ and, by the induction hypothesis, $T^{\prime}\in\mathcal{T}$. Notice that $v_{3}$ is not a quasi-isolated vertex of $T^{\prime}$, otherwise Lemma 9 would imply $\gamma_{t}(T)=\gamma_{t}(T^{\prime})+1$, but $\gamma_{t}(T^{\prime})=\gamma(T)-2$. Therefore, $v_{3}$ is not a quasi-isolated vertex of $T^{\prime}$, and we can obtain $T$ from $T^{\prime}$ using operation $\mathcal{O}_{4}$ and $T\in\mathcal{T}$.

Now we may assume that $d_{T}(v_{2})=2$. For purposes of contradiction, suppose that $d_{T}(v_{3})>2$. Hence, there is a path $P_{3}=(a,b,c)$ which is attached to $v_{3}$ by the edge $cv_{3}$. Since $D$ is a $\gamma_{t}(T)$-set, we have $b,c\in D$. But then $(D\cup\{v_{3}\})\setminus\{v_{2},c\}$ is a vertex cover of $T,$ a contradiction. Thus $d_{T}(v_{3})=2$.

Since $D$ is a vertex cover of $T$ and $v_{3}\notin D,$ we have $v_{4}\in D$. If $d_{T}(v_{4})=1$, then $T=P_{5}$, and $D$ is not a $\gamma_{t}(T)$-set. If $d_{T}(v_{4})=2$, then $v_{5}\in D$, in this case $(D\setminus\{v_{2},v_{4}\})\cup\{v_{3}\}$ is a vertex cover of $T$, a contradiction. Hence, $d_{T}(v_{4})>2.$

Define $T^{\prime}$ as $T^{\prime}=T-\{v_{0},v_{1},v_{2},v_{3}\}$, the set $D^{\prime}=D\setminus\{v_{1},v_{2}\}$ is a $(\gamma_{t}-\tau)$-set of $T^{\prime}$ containing $v_{4}$, and by the induction hypothesis, $T^{\prime}\in\mathcal{T}$. Thus, we can obtain $T$ from $T^{\prime}$ using operation $\mathcal{O}_{1}$ on $T\in\mathcal{T}$.

∎

Therefore, we have proved the following theorem.

Theorem 11**.**

It $T$ is a tree, then $T\in\mathcal{T}$ if and only if $T$ has a $(\gamma_{t}-\tau)$-set.

4 Further work and open problems

Once we have characterized the trees having a $(\gamma_{t}-\tau)$-set, the following natural step is to consider the following problem.

Problem 12**.**

Find a characterization for the ($\gamma_{t}-\tau$)-trees.

If we let $\mathcal{T}^{\prime}$ be the family of all $(\gamma_{t}-\tau)$-trees, it is clear that the family $\mathcal{T}$, of all trees having a $(\gamma_{t}-\tau)$-set, is contained in $\mathcal{T}^{\prime}$. We have already observed in Section 2, that this containment is proper. Moreover, we can slightly modify the operations $\mathcal{O}_{1},\mathcal{O}_{2}$, and $\mathcal{O}_{3}$ to preserve the equality $\gamma_{t}=\tau$, but not necessarily preserving the existence of a $(\gamma_{t}-\tau)$-set, thus obtaining a larger infinite family of trees, say $\mathcal{S}$, such that $\mathcal{T}\subset\mathcal{S}\subset\mathcal{T}^{\prime}$. The modified operations for a tree $T$ are the following (notice the relaxation of the choice of $u$, cf. Section 2).

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Operation $\mathcal{O}^{\prime}_{1}$: Let $u$ be a vertex in $T$, and let $v$ be a leaf of a path $P_{4}$. Then do the sum of $T$ with $P_{4}$ via the edge $uv$.

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Operation $\mathcal{O}^{\prime}_{2}$: Let $u\in V(T)$ such that $u$ belongs to some $\gamma_{t}(T)$-set and also belongs to some $\tau(T)$-set. Then add a new vertex $v$ to $T$ supported by $u$.

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Operation $\mathcal{O}^{\prime}_{3}$: Let $u\in V(T)$ such that $u$ belongs to some $\gamma_{t}(T)$-set and $u$ it is not a quasi-isolated vertex. Let $P_{2}=(v,w)$ be a path with two vertices. Then do the sum of $T$ with $P_{2}$ via the edge $uv$.

Notice that the family of paths of length $4k$, $k\geq 2$, mentioned in Section 2 as an example of an infinite family of $(\gamma_{t}-\tau)$-graphs not having a $(\gamma_{t}-\tau)$-set, can be obtained from $P_{4}$ by recursively applying operation $\mathcal{O}^{\prime}_{1}$; this shows that the inclusion $\mathcal{T}\subset\mathcal{T}^{\prime}$ is proper. Similarly, examples can be found of a tree $T^{\prime}$ obtained from a tree $T$ by applying operation $\mathcal{O}_{i}$, $i\in\{2,3\}$, such that $T$ has a $(\gamma_{t}-\tau)$-set, but $T^{\prime}$ does not.

Thus, the family $\mathcal{S}$ above defined is a good starting point to look for the class of all $(\gamma_{t}-\tau)$-trees. It is worth noticing that there are many ad-hoc operations that could be defined, both on trees and general graphs, that preserve the equality $\gamma_{t}=\tau$ (e.g., subdividing an edge four times). Nonetheless, there is no obvious choice for a set of operations similar to the one used to prove Theorem 11, that will lead to a solution for Problem 12. Maybe, instead of a characterization using a set of operations, the following idea could be useful. Consider two $(\gamma_{t}-\tau)$-graphs, $G_{i}=(V_{i},E_{i})$, and $X_{i}\subseteq V_{i}$, $i\in\{1,2\}$, we want to add some edges joining the vertices of $X_{1}$ with the vertices of $X_{2}$ so that the resulting graph is also a $(\gamma_{t}-\tau)$-graph. What conditions do we need to achieve this goal? Consider the following two examples. First, if $G_{1}$ is an empty graph, $|V_{1}|=|V_{2}|$, $X_{i}=V_{i}$, for $i\in\{1,2\}$, and we add a perfect matching between $X_{1}$ and $X_{2}$, we obtain the corona of the graph $G_{2}$, which is a $(\gamma_{t}-\tau)$-graph. Second, if $G_{1}$ is a $P_{4}$, $X_{1}$ is a singleton containing an end-vertex of $G_{1}$, and $X_{2}$ is a singleton containing any vertex of $G_{2}$, then we are describing operation $\mathcal{O}^{\prime}_{1}$, and again, the resulting graph is a $(\gamma_{t}-\tau)$-graph. These two “extremal” cases, where $X_{i}$, $i\in\{1,2\}$ has the largest and smallest possible cardinalities, respectively, seem to be the easiest to handle. So, another kind of recursive characterization could be obtained if, for example, one could prove that every $(\gamma_{t}-\tau)$-graph could be obtained by the sum via and edge, from two smaller $(\gamma_{t}-\tau)$-graphs, or by adding a perfect matching between two smaller $(\gamma_{t}-\tau)$-graphs.

From the computational point of view, for any tree $T$, both $\gamma_{t}(T)$ and $\tau(T)$ can be determined in polynomial time. Hence, the problem of determining if $\gamma_{t}(T)=\tau(T)$, for a tree $T$, is polynomial time solvable. For the case of trees having a $(\gamma_{t}-\tau)$-set, Theorem 11 does not trivially imply a polynomial algorithm to determine the existence of a $(\gamma_{t}-\tau)$-set in a tree, so the following problem seems to be interesting.

Problem 13**.**

Find the complexity of determining the existence of a $(\gamma_{t}-\tau)$-set in a tree.

Of course, it is also interesting to ask both problems for general graphs.

Problem 14**.**

For a given graph $G$:

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Find the complexity of determining whether $\gamma_{t}(G)=\tau(G)$.

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Find the complexity of determining the existence of a $(\gamma_{t}-\tau)$-set in $G$.

Our intuition says that the existence of a $(\gamma_{t}-\tau)$-set is so restrictive in the structure of $G$ that the second problem might be solved in polynomial time.