Hinge Total Domination on Some Graph Families

Abstract

set S of vertices in a graph G = (V (G);E(G)) is a hinge dominating set if every vertex \(u\) \(\in\) V \(\setminus \) \(S\) is adjacent to some vertex \(u\) \(\in\) \(S\) and a vertex \(w\) \(\in\) V \(\setminus\) \(S\)  such that (\(v\), \(w\)) is not an edge in E(G). The hinge domination number \(\gamma\)\(h\)(\(G\)) is the minimum size of a hinged dominating set. A set S is called a total dominating set of G if for every vertex in V , including those in S is adjacent to at least one vertex in S. The cardinality of a minimum total dominating set in G is called the total domination number of G and denoted as \(\gamma\)\(h\)(\(G\)) In this study, a new parameter called hinged total dominating set was introduced and defined as, a hinge total dominating set of a graph G is a set S of vertices of G such that S is both a hinge dominating set and total dominating set. The hinge total domination number, \(\gamma\)\(h\)(\(G\)) is the minimum cardinality of a hinge total dominating set of G. We initiate a study of hinge total dominating set and present its characterization. In addition, we also determine the exact values of hinge total domination number on some graph families.