Total double Roman domination numbers in digraphs

Let [Formula: see text] be a finite and simple digraph with vertex set [Formula: see text]. A double Roman dominating function (DRDF) on digraph [Formula: see text] is a function [Formula: see text] such that every vertex with label 0 has an in-neighbor with label 3 or two in-neighbors with label 2 and every vertex with label 1 have at least one in-neighbor with label at least 2. The weight of a DRDF [Formula: see text] is the value [Formula: see text]. A DRDF [Formula: see text] on [Formula: see text] with no isolated vertex is called a total double Roman dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertex. In this paper, we initiate the study of the total double Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some bounds for the total double Roman domination number and we determine the total double Roman domination number of some classes of digraphs.

Tight Bounds on the Clique Chromatic Number

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

Resolving Restrained Domination in Graphs

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs

Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of v∈V(G). Let f:V(G)→{0,1,2} be a function and Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V1∪V2 has no isolated vertex and N(v)∩V2≠∅ for every v∈V(G)\V2. The strongly total Roman domination number of G, denoted by γtRs(G), is defined as the minimum weight ω(f)=∑x∈V(G)f(x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing γtRs(G) is NP-hard.

Total Roman 2 -Reinforcement of Graphs

A total Roman 2 -dominating function (TR2DF) on a graph Γ = V , E is a function l : V ⟶ 0,1,2 , satisfying the conditions that (i) for every vertex y ∈ V with l y = 0 , either y is adjacent to a vertex labeled 2 under l , or y is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF l is the value ∑ y ∈ V l y . The total Roman 2 -domination number (TR2D-number) of a graph Γ is the minimum weight of a TR2DF on Γ . The total Roman 2 -reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs.

The Irregularity and Modular Irregularity Strength of Fan Graphs

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

A Note on the Paired-Domination Subdivision Number of Trees

For a graph G with no isolated vertex, let γpr(G) and sdγpr(G) denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order n≥4 different from a healthy spider (subdivided star), then sdγpr(T)≤min{γpr(T)2+1,n2}, improving the (n−1)-upper bound that was recently proven.

Total pitchfork domination and its inverse in graphs

New two domination types are introduced in this paper. Let [Formula: see text] be a finite, simple, and undirected graph without isolated vertex. A dominating subset [Formula: see text] is a total pitchfork dominating set if [Formula: see text] for every [Formula: see text] and [Formula: see text] has no isolated vertex. [Formula: see text] is an inverse total pitchfork dominating set if [Formula: see text] is a total pitchfork dominating set of [Formula: see text]. The cardinality of a minimum (inverse) total pitchfork dominating set is the (inverse) total pitchfork domination number ([Formula: see text]) [Formula: see text]. Some properties and bounds are studied associated with maximum degree, minimum degree, order, and size of the graph. These modified domination parameters are applied on some standard and complement graphs.

On the Outer-Independent Roman Domination in Graphs

Let G be a graph with no isolated vertex and f:V(G)→{0,1,2} a function. Let Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. The function f is an outer-independent Roman dominating function on G if V0 is an independent set and every vertex in V0 is adjacent to at least one vertex in V2. The minimum weight ω(f)=∑v∈V(G)f(v) among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs.

Total Roman Domination Number of Rooted Product Graphs

Let G be a graph with no isolated vertex and f:V(G)→{0,1,2} a function. If f satisfies that every vertex in the set {v∈V(G):f(v)=0} is adjacent to at least one vertex in the set {v∈V(G):f(v)=2}, and if the subgraph induced by the set {v∈V(G):f(v)≥1} has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight ω(f)=∑v∈V(G)f(v) among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.