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 ?a. A vertex u strongly (weakly) a-dominates a vertex ? if, deg<inf>a</inf> u ? deg<inf>a</inf> ? (deg<inf>a</inf> u ? deg<inf>a</inf> ?) where deg<inf>a</inf> u is the number of vertices adjacent to u. A set D ? V(G) is a strong (weak) adset [sad-set (wad-set)], if every vertex in V - D is strongly (weakly) a-dominated by at least one vertex in D. This paper presents some new results on strong (weak) domination in semigraphs. © 2005 Elsevier Ltd. All rights reserved.

Kamath, S.S.Hebbar, S.R.2026-02-05Strong andElectronic Notes in Discrete Mathematics, 2003, 15, , pp. 112-15710653https://doi.org/10.1016/S1571-0653(04)00549-9https://idr.nitk.ac.in/handle/123456789/279772003Sampathkumar [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 ?a. A vertex u strongly (weakly) a-dominates a vertex ? if, deg<inf>a</inf> u ? deg<inf>a</inf> ? (deg<inf>a</inf> u ? deg<inf>a</inf> ?) where deg<inf>a</inf> u is the number of vertices adjacent to u. A set D ? V(G) is a strong (weak) adset [sad-set (wad-set)], if every vertex in V - D is strongly (weakly) a-dominated by at least one vertex in D. This paper presents some new results on strong (weak) domination in semigraphs. © 2005 Elsevier Ltd. All rights reserved.