Bhat, R S and Kamath , S S and Bhat, Surekha R (2011) A Bound on Weak Domination Number Using Strong (Weak) Degree Concepts in Graphs. Journal of International Academy of Physical Sciences, 15 (3). pp. 303-317. ISSN 0974 - 9373

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Abstract

For a vertex v of a graph G = (V, X ), N(v)={uV| u is adjacent to v}. Then degree of the vertex v, d(v) = | N(v) | . We define Ns(v)={u N(v) !d (u}. Similarly, Nw(v)+{u N(v)!d(v) <d(u)} . Then Strong degree of a vertex v is dw (v)=Nw(v)!. and weak degree of a vertex v is dw(v)=!Nw(v)!. Consequently, we have the following graph parameters – maximum strong degree (G) minimum strong degree w(G) maximum weak degree and minimum weak degree w(g) For any two adjacent vertices u and v in a graph G (V, X ), u strongly [weakly] dominates v if d(u)>d(v) [d(u)<d(v)]. A set D V is a dominating set (strong dominating set [sd-set], weak dominating set [wd-set] respectively) of G if every v V-D is dominated (strongly dominated, weakly dominated respectively) by some u D. The domination number (strong domination number, weak domination number respectively) y-y(G) (Ys=Ys(G) Yw=Yw(G) respectively) of G is the minimum cardinality of a dominating set (sd- set, wd-set respectively) of G. In this paper, we obtain a new lower bound and also an improved upper bound for the weak domination number using the new parameters. A new version of the “First Theorem in Graph Theory” is presented. We define a new matrix called strong weak adjacency matrix of a graph and give an algorithm to obtain the strong weak adjacency matrix from the adjacency matrix and vice versa. Using the properties of the new matrix, we can compute all the information about the new parameters defined above.

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