Dominating cliques in graphs

AbstractA set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique. A forbidden subgraph characterization is given for a class of graphs that have a connected dominating set of size three

d-Lucky Labeling of Graphs

AbstractLet l: V (G) →N be a labeling of the vertices of a graph G by positive integers. Define , where d(u) denotes the degree of u and N(u) denotes the open neighborhood of u. In this paper we introduce a new labeling called d-lucky labeling and study the same as a vertex coloring problem. We define a labeling l as d-lucky if c(u) ≠ c(v) , for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted by ηdl(G), is the least positive k such that G has a d-lucky labeling with {1,2, ..., k} as the set of labels. We obtain ηdl(G) = 2 for hypercube network, butterfly network, benes network, mesh network, hypertree and X-tree

The complexity of connected domination and total domination by restricted induced graphs

Given a graph class C, it is natural to ask whether a given graph has a connected or a total dominating set inducing a graph of C and, if so, what is the minimal size of such a set. We give a sufficient condition on C for the intractability of this problem. This condition is fulfilled by a wide range of graph classes

The complexity of connected domination and total domination by restricted induced graphs

Given a graph class C, it is natural to ask whether a given graph has a connected or a total dominating set inducing a graph of C and, if so, what is the minimal size of such a set. We give a sufficient condition on C for the intractability of this problem. This condition is fulfilled by a wide range of graph classes

On the existence of total dominating subgraphs with a prescribed additive hereditary property

AbstractRecently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set

A Graph Class related to the Structural Domination Problem

In the structural domination problem one is concerned with the question if a given graph has a connected dominating set whose induced subgraph has certain structural properties. For most of the common graph properties, the associated decision problem is NP-hard. Recently, Bacsô and Tuza gave a full characterization of the graphs whose every induced subgraph has a connected dominating set satisfying an arbitrary prescribed hereditary property. Using the Theorem of Bacsô and Tuza, we derive a finite forbidden subgraph characterization of the distance-hereditary graphs that have a dominating induced tree. Furthermore, we show that for distance-hereditary graphs minimum dominating induced trees can be found efficiently. The main part of the paper studies a new class of graphs, the extit{structural domination class}, which is closely related to the structural domination problem. Among other results, we give new characterizations of certain subclasses of distance-hereditary graphs (in particular for ptolemaic graphs) and analyse the structure of minimum connected dominating sets of structural domination graphs. It turns out that many of the problems associated to structural domination become tractable on the hereditary part of the structural domination class

A Graph Class related to the Structural Domination Problem

In the structural domination problem one is concerned with the question if a given graph has a connected dominating set whose induced subgraph has certain structural properties. For most of the common graph properties, the associated decision problem is NP-hard. Recently, Bacsô and Tuza gave a full characterization of the graphs whose every induced subgraph has a connected dominating set satisfying an arbitrary prescribed hereditary property. Using the Theorem of Bacsô and Tuza, we derive a finite forbidden subgraph characterization of the distance-hereditary graphs that have a dominating induced tree. Furthermore, we show that for distance-hereditary graphs minimum dominating induced trees can be found efficiently. The main part of the paper studies a new class of graphs, the extit{structural domination class}, which is closely related to the structural domination problem. Among other results, we give new characterizations of certain subclasses of distance-hereditary graphs (in particular for ptolemaic graphs) and analyse the structure of minimum connected dominating sets of structural domination graphs. It turns out that many of the problems associated to structural domination become tractable on the hereditary part of the structural domination class

Domination on hyperbolic graphs

If k ≥ 1 and G = (V, E) is a finite connected graph, S ⊆ V is said a distance k-dominating set if every vertex v ∈ V is within distance k from some vertex of S. The distance k-domination number γ kw (G) is the minimum cardinality among all distance k-dominating sets of G. A set S ⊆ V is a total dominating set if every vertex v ∈ V satisfies δS (v) ≥ 1 and the total domination number, denoted by γt(G), is the minimum cardinality among all total dominating sets of G. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain relationships between the hyperbolicity constant δ(G) and some domination parameters of a graph G. The results in this work are inequalities, such as γkw(G) ≥ 2δ(G)/(2k + 1) and δ(G) ≤ γt(G)/2 + 3.Supported by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain, and a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain