Domination Critical Semigraphs

doi:10.1016/S1571-0653(04)00550-5

Electronic Notes in Discrete Mathematics

Volume 15, May 2003, Page 113

https://doi.org/10.1016/S1571-0653(04)00550-5 Get rights and content

Abstract

Sampathkumar [1] introduced a new type of generalization to graphs, called Semigraphs. A semigraph G = (V, X) on the set of vertices V and the set of edges X consists of n-tuples (u₁,u₂,u_n) of distinct elements belonging to the set V for various n ≥ 2, with the following conditions : (1) Any n-tuple (u₁,u₂,u_n) = (u_n,u_n-1,…,u₁) and (2) Any two such tuples have at most one element in common.

S. S. Kamath and R. S. Bhat [3] introduced domination in semigraphs. Two vertices u and v are said to a-dominate each other if they are adjacent. A set D ⊆ V(G) is an adjacent dominating set (ad-set) if every vertex in V - D is adjacent to a vertex in D. The minimum cardinality of an ad-set D is called adjacency domination number of G and is denoted by γ_a.

γ_a(G) may increase or decrease by the removal of a vertex or an edge from G.

A vertex v of a semigraph G is said to be γ_a - critical if γ_a(G - v) ≠ γ_a(G); if γ_a(G - v) = γ_a(G), then v is γ_a - redundatnt. The main objective of this paper is to study this phenomenon on the vertices and edges of a semigraph.

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Citation Excerpt :
Some results on the energy of a semigraph, making use of the adjacency matrix associated with a semigraph, are studied in [12]. The domination numbers associated with semigraphs are investigated in [13–16]. In a semigraph, an edge of size at least 3 is known as a semiedge.
E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete.
Let P be a set with p elements and ${E_{1}, E_{2}, \dots, E_{q}}$ be a collection of subsets of P with $⋃_{i = 1}^{q} E_{i} = P$ . We derive an expression for the maximum value of the difference $(\sum_{j = 1}^{k} | E_{i_{j}} | - |⋃_{i = 1}^{k} E_{i_{j}}|)$ for $2 ⩽ k ⩽ q$ , where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges ${e_{1}, e_{2}, \dots, e_{q}}$ , where the set $E_{i}$ is the set of vertices on the edge $e_{i}, 1 ⩽ i ⩽ q$ . Combining the above two results, we characterize a complete semigraph.
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