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 (u<inf>1</inf>,u<inf>2</inf>,u<inf>n</inf>) of distinct elements belonging to the set V for various n ? 2, with the following conditions : (1) Any n-tuple (u<inf>1</inf>,u<inf>2</inf>,u<inf>n</inf>) = (u<inf>n</inf>,u<inf>n-1</inf>,...,u<inf>1</inf>) 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 ?<inf>a.</inf>. ?<inf>a</inf>(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 ?<inf>a</inf> - critical if ?<inf>a</inf>(G - v) ? ?<inf>a</inf>(G); if ?<inf>a</inf>(G - v) = ?<inf>a</inf>(G), then v is ?<inf>a</inf> - redundatnt. The main objective of this paper is to study this phenomenon on the vertices and edges of a semigraph. © 2005 Elsevier Ltd. All rights reserved.

Kamath, S.S.Hebbar, S.R.2026-02-05DominationElectronic Notes in Discrete Mathematics, 2003, 15, , pp. 113-15710653https://doi.org/10.1016/S1571-0653(04)00550-5https://idr.nitk.ac.in/handle/123456789/279792003Sampathkumar [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 (u<inf>1</inf>,u<inf>2</inf>,u<inf>n</inf>) of distinct elements belonging to the set V for various n ? 2, with the following conditions : (1) Any n-tuple (u<inf>1</inf>,u<inf>2</inf>,u<inf>n</inf>) = (u<inf>n</inf>,u<inf>n-1</inf>,...,u<inf>1</inf>) 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 ?<inf>a.</inf>. ?<inf>a</inf>(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 ?<inf>a</inf> - critical if ?<inf>a</inf>(G - v) ? ?<inf>a</inf>(G); if ?<inf>a</inf>(G - v) = ?<inf>a</inf>(G), then v is ?<inf>a</inf> - redundatnt. The main objective of this paper is to study this phenomenon on the vertices and edges of a semigraph. © 2005 Elsevier Ltd. All rights reserved.