Gallai-type theorems and domination parameters - Science Direct
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Gallai-type theorems and domination parameters - Science Direct
aDepartment of Mathematics and Computer Science, Georgia State University, Atlanta, .... Let G be a graph with i(G) + A(G) = n and let x be a vertex of degree.
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DISCRETE MATHEMATICS ELSEVIER
Discrete Mathematics 167/168 (1997) 237-248
Gallai-type theorems and domination parameters Gayla S. Domke a, Jean E. Dunbar b, Lisa R. Markus c,* aDepartment of Mathematics and Computer Science, Georgia State University, Atlanta, GA 30303, USA bDepartment of Mathematics, Computer Science and Physics, Converse College, Spartanburg, SC 29302, USA CDepartment of Mathematics, Furman University, Greenville, SC 29613, USA Received 7 July 1995; revised 5 February 1996
Abstract
Let 7(G) denote the minimum cardinality of a dominating set of a graph G = (V,E). A longstanding upper bound for 7(G) is attributed to Berge: For any graph G with n vertices and maximum degree A(G), 7(G) ~ n - 1.
Theorem 14. L e t G be a connected graph, and let x be a vertex o f degree A(G). I f V - Nix] is an independent set and every vertex in N ( x ) is adjacent to at most one vertex in V - N[x], then either 7(G) + A ( G ) = n or 7(G) + A ( G ) = n - I.
Proof. Suppose G is a connected graph and x that V - N [ x ] is an independent set and every vertex in V - N[x]. Any minimal dominating vertices in order to dominate all the vertices by using x and all o f V - X [ x ] . So we have Thus n - A ( G ) - 1 ~< 7(G)
