On graphs all of whose total dominating sequences have the same length

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Abstract

A sequence of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex v in the sequence has a neighbor which is adjacent to no vertex preceding v in the sequence, and at the end every vertex of G has at least one neighbor in the sequence. Minimum and maximum lengths of a total dominating sequence is the total domination number of G (denoted by γt(G)) and the Grundy total domination number of G (denoted by γgrt(G)), respectively. In this paper, we study graphs where all total dominating sequences have the same length. For every positive integer k, we call G a total k-uniform graph if every total dominating sequence of G is of length k, that is, γt(G)=γgrt(G)=k. We prove that there is no total k-uniform graph when k is odd. In addition, we present a total 4-uniform graph which stands as a counterexample for a conjecture by [11] and provide a connected total 8-uniform graph. Moreover, we prove that every total k-uniform, connected and false twin-free graph is regular for every even k. We also show that there is no total k-uniform chordal connected graph with k4 and characterize all total k-uniform chordal graphs.

Introduction

Let G be a simple graph with vertex set V(G) and edge set E(G). The neighborhood of a vertex vV(G), denoted by N(v), is the set of vertices adjacent to v. The closed neighborhood of a vertex vV(G), denoted by N[v], is N(v){v}. A subset A of V(G) is called a dominating set of G if every vertex in V(G)A has at least one neighbor in A. If G has no isolated vertices, a subset AV(G) is called a total dominating set of G if every vertex of V(G) is adjacent to at least one member of A. The total domination number of G with no isolated vertices, denoted by γt(G), is the minimum size of a total dominating set of G.

A sequence S=(v1,,vk) of distinct vertices of G is a legal (open neighborhood) sequence ifN(vi)j=1i1N(vj) holds for every i{2,,k}. If, in addition, {v1,,vk} is a total dominating set of G, then we call S a total dominating sequence of G. The maximum length of a total dominating sequence in G is called the Grundy total domination number of G and it is denoted by γgrt(G).

It is clear that the length of a total dominating sequence in G is at least γt(G), and any permutation of a minimum total dominating set of G forms a total dominating sequence attaining this lower bound. In this paper, we study graphs in which the Grundy total domination number is equal to the total domination number. A graph G is called total k-uniform if γt(G)=γgrt(G)=k. In other words, a graph G is total k-uniform if and only if every total dominating sequence is of length k. Total k-uniform graphs are indeed total domination version of k-uniform graphs introduced in [10]. A sequence (v1,,vk) is called dominating closed neighborhood sequence if {v1,,vk} is a dominating set and N[vi]j=1i1N[vj] holds for every i{2,,k}. A graph is called k-uniform whenever every dominating closed neighborhood sequence is of length k. k-uniform graphs with k3 are characterized in [6], whereas the work in [10] provided the complete characterization of k-uniform graphs.

Numerous variants of Grundy total domination such as Grundy domination, Z-Grundy domination and L-Grundy domination exist in the literature [6], [5], [3], [4]. The work in [4] investigated the relations between these types of Grundy domination as well as the relation between the Z-Grundy domination number and the zero forcing number of a graph.

The concept of total domination in graphs was introduced in 1980 [9] and has been studied extensively in the literature (see [13]). The parameter γgrt(G) was first introduced by [7], who obtained bounds on γgrt(G) for trees and regular graphs in terms of other graph variants. The decision versions of both the total domination number and the Grundy total domination number are NP-complete [7]. Indeed, the problem of finding the Grundy total domination number is NP-hard in bipartite graphs [7] and split graphs [8], while it is solvable in polynomial time in trees, bipartite distance-hereditary graphs, and P4-tidy graphs [8].

Various bounds for Grundy total domination number have been obtained for several graph classes such as regular graphs and graph products [2], [7]. The work in [7] also characterized the graphs where the Grundy total domination number attains its trivial upper bound |V(G)|. In their work, it is additionally shown that complete multipartite graphs are the only total 2-uniform graphs and there are no total 3-uniform graphs. In this paper, we generalize their result for total 3-uniform graphs and prove that there does not exist any total k-uniform graph when k is odd and hence, we partially solve the open problem (characterizing the graphs G such that γt(G)=γgrt(G)=k for k4) posed by [7]. We also obtain that removing all vertices with the same neighborhood but except one from a connected total k-uniform graph gives rise to a regular graph.

Paper [11] characterized total 4-uniform bipartite graphs, showed that there is no connected total 4-uniform chordal graph and established a correspondence between regular total 6-uniform bipartite graphs and some certain finite projective planes. The authors also claimed that any connected total 4-uniform graph is bipartite. In this paper, we disprove their conjecture by exhibiting a graph with fifteen vertices. We additionally prove that there are no connected total k-uniform chordal graphs when k4; hence, we classify all total k-uniform chordal graphs.

The remainder of this paper is organized as follows. In Section 2 we provide a reduction from total k-uniform graphs to total (k2)-uniform graphs, which is essential to this paper and implies that there does not exist a total k-uniform graph when k is odd. In Section 3, we present a connected, non-bipartite and total 4-uniform graph together with a connected total 8-uniform graph. In Section 4, we show that connected, false twin-free and total k-uniform graphs are regular. Section 5 contains the complete characterization of total k-uniform chordal graphs. Discussion and conclusions are provided in Section 6.

Section snippets

A reduction from total k-uniform graphs to total (k-2)-uniform graphs

If G is a graph with no isolated vertices, then every legal sequence in G which is not a total dominating sequence can be extended to a total dominating sequence of G. Moreover, a graph G contains a total dominating sequence if and only if G has no isolated vertices. We shall implicitly make use of these observations in the sequel.

A graph all of whose minimal total dominating sets are of the same size is called a well-totally-dominated graph (see [1]). Well-totally-dominated graphs are

Total k-uniform graphs with small k

In this section, we introduce a new connected total 4-uniform graph and based on this graph, we present a new connected total 8-uniform graph.

The graph Kn,nM, where M is an arbitrary perfect matching of Kn,n, is called a crown graph on 2n vertices. Graphs which are isomorphic to KnKm, where m,n2, or a crown graph on at least 6 vertices are members of G4. It is easy to see that the former ones are the only disconnected graphs in G4. The work in [11] showed that the latter ones are the only

Regularity

In this section, for every even positive integer k we show that any connected graph in Gk is regular. We start with studying bipartite graphs.

Proposition 4.1

Let G be a bipartite, connected, false twin-free and total k-uniform graph for some even positive integer k. Then, G is a regular graph.

Proof

The proof is by strong induction on k. For k=2, such a graph is K2 and trivially it is regular. Now let k4 be an even integer and assume that the statement is true for every positive even k less than k.

Let X and Y be

Total k-uniform chordal graphs

In this section, we characterize all total k-uniform chordal graphs. In [11] Theorem 3.3 states that there is no connected chordal graph G with γt(G)=γgrt(G)=4. We extend their result and show that there is no connected total k-uniform chordal graph when k4. Notice that to show this result, it suffices to prove that Gk has no connected chordal graph for k4.

Lemma 5.1

If G is a connected graph in Gk where k4, then G has an induced C5 or C6.

Proof

Let G be a connected graph in Gk with k4. It is well-known

Discussion and conclusions

Recall that for every integer l satisfying γt(G)lγgrt(G) there exists a total dominating sequence of length l in G ([6], [7]). Therefore, a graph is total k-uniform if and only if G has at least one total dominating sequence of length k but has no total dominating sequence of length k1 or k+1. Thus, for a fixed positive integer k, the problem of determining whether a given graph is total k-uniform is clearly solvable in polynomial time. On the other hand, both of the problems of finding the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under grant no. 118E799. The work of Didem Gözüpek is also supported by the BAGEP Award of the Science Academy of Turkey. We would like to thank Douglas Rall for making us aware of some properties of the direct product of graphs in Section 3.

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