S.Kiruthiga Deepa. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 9, ( Part -4) September 2017, pp.61-65 www.ijera.com DOI: 10.9790/9622-0709046165 61 | Page Strong and Weak Vertex-Edge Mixed Domination on S - Valued Graphs S.Kiruthiga Deepa * and M.Chandramouleeswaran ** * Bharath Niketan College of Engineering Aundipatti - 625536. Tamilnadu. India. ** Saiva Bhanu Kshatriya College Aruppukottai - 626101. Tamilnadu. India. Corresponding author: S.Kiruthiga Deepa ABSTRACT In [6] we have introduced the notion of vertex edge mixed domination in S-valued graphs and proved several results. In this paper we introduce the notion of strong and weak vertex - edge mixed domination on S-valued graphs. MS Classification: 05C25, 16Y60 Keywords: S-valued graphs, strong ve-weight m-dominating set, weak ve-weight m-dominating set. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 04-09-2017 Date of acceptance: 13-09-2017 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION Motivated by the definition of S-valued graph [2], Chandramouleeswaran and others [9] introduced the notion of semi ring valued graphs. In [5] and [8] the authors discussed regularity conditions on S-valued graphs. In [7] we have defined degree regularity on edges of S-valued graphs. The theory of domination was initiated by Berge [1] and was studied by many researchers. Motivated by the works on domination in crisp graph theory in [3] and [4] the authors introduced the notion of vertex domination and strong-weak vertex domination on S-valued graphs respectively. In [6] we have introduced the notion of vertex-edge mixed domination in S-valued graphs and proved several results. In this paper we introduce the notion of strong and weak vertex - edge mixed domination on S-valued graphs. II. PRELIMINARIES In this section, we recall some basic definitions that are needed for our work. Definition 2.1: [2] A semi ring (S, +, .) is an algebraic system with a non-empty set S together with two binary operations + and . such that (1) (S, +, 0) is a monoid. (2) (S, . ) is a semigroup. (3) For all a, b, c  S , a . (b + c) = a . b + a . c and (a + b) . c = a . c + b . c (4) 0 . x = x . 0 = 0 ,  x  S. Definition 2.2:[2] Let (S, +, .) be a semiring. A Canonical Pre-order  in S defined as follows: for a, b  S , a  b if and only if, there exists an element c  S such that a + c = b. Definition 2.3: [1] A subset D  V of vertices in a graph G = (V, E) is called a vertex dominating set in G if every vertex v  V is either an element in D or is adjacent to an element in V – D. A subset D  V is a vertex dominating set of G, if  v  V – D,    D v N  Definition 2.4: [1] A subset D  V is an independent set of G, if u, v  D,     v u N  Definition 2.5: [1] A subset D  V is an Independent dominating set of G if D is both an independent and a dominating set. Definition 2.6: [9] Let G = (V, E  V  V ) be a given graph with V, E   . For any semiring (S, +, .), a semi ring-valued graph (or a S-valued graph), G S , is defined to be the graph ) , , , (   E V G S  where S V  :  and S E  :  is defined to be       otherwise x y or y x if y x y x , 0 ) ( ) ( ) ( ) ( )}, ( ), ( min{ ,          For every unordered pair (x,y) of E  V  V. We call  , a S- vertex set and ,  a S-edge set of G S . Definition 2.7: [9] If  (v) = a;  v  V and some a  S then the corresponding S-valued graph G S is called a vertex regular S-valued graph. Definition 2.8: [3] Consider the S-valued graph G S =(V, E  V  V ). The open neighbourhood of v i in G S is defined as the set     . ) , ( , ) , ( ) ( , ) ( S v v E v v where v v v N j i j i j j i S      RESEARCH ARTICLE OPEN ACCESS