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Estimations for the domination number of a graph

โœ Scribed by Peter Flach; Lutz Volkmann


Publisher
Elsevier Science
Year
1990
Tongue
English
Weight
345 KB
Volume
80
Category
Article
ISSN
0012-365X

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For r > 0, let the r-domination number of a graph, d,, be the size of a smallest set of vertices such that every vertex of the graph is within distance r of a vertex in that set. This paper contains proofs that every graph with a spanning tree with at least n/2 leaves has d, s n/(2r); this compares

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The kdomination number of a graph G, y k ( G ) , is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k. then YAG) 5 kp/(k + 1).

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Topp, J. and L. Volkmann, Some upper bounds for the product of the domination number and the chromatic number of a graph, Discrete Mathematics 118 (1993) 2899292. Some new upper bounds for yx are proved, where y is the domination number and x is the chromatic number of a graph. All graphs consider

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For a graph G, the definitions of doknation number, denoted y(G), and independent domination number, denoted i(G), are given, and the following results are obtained: oorollrrg 1. For any graph G, y(L(G)) = i@(G)), where Z,(G) is the line graph of G. (This $xh!s t.lic rtsult ~(L(T))~i(L(T)), h w ere

Domination numbers of planar graphs
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The problem of determining the domination number of a graph is a well known NPhard problem, even when restricted to planar graphs. By adding a further restriction on the diameter of the graph, we prove that planar graphs with diameter two and three have bounded domination numbers. This implies that