On secure domination number of generalized Mycielskian of some graphs

Abstract

Generalized Mycielskians are triangle-free networks with a large chromatic number, having a number of desirable characteristics like fast multi-path communication, high fault tolerance, reliable resource sharing, etc. Graph invariants like domination number and secure domination number can be used to protect the network by monitoring each node or moving to a failing node. A dominating set of a graph G is a subset of its vertex set, which can monitor every other vertex of the graph, and γ (G), the domination number of G, is the least cardinality among all dominating sets of G. A secure dominating set S of G is a dominating set with an additional property that, for each vertex u not in S there is a vertex v in S adjacent to u such that, the swapped set (S - {v}) ∪ {u} is dominating. γs (G) is the secure domination number of G, which is the minimum cardinality among all secure dominating sets of G. In this paper, we analyzed γ and γs of the generalized Mycielskian (μm) for path graphs Pn and cycle graphs Cn, varying both n and m, and used these results to obtain bounds for general graphs G.