Paths, stars and the number three.(English)Zbl 0857.05052

A dominating set for a graph $$G$$ is a set $$D$$ of vertices of $$G$$ such that every vertex in $$V(G)-D$$ has a neighbour in $$D$$. In this paper it is proved that any graph $$G$$ of order $$n$$ with minimum degree at least 3 has a dominating set of size at most $$3n/8$$. This is the same as saying that the graph can be covered by at most $$3n/8$$ stars (a star $$S$$ being a vertex $$v$$ with edges to all vertices in $$S-v$$). It is also shown that any connected cubic graph $$G$$ of order $$n$$ can be covered by $$\lceil n/9\rceil$$ vertex disjoint paths. Both of these results are sharp.

MSC:

 05C35 Extremal problems in graph theory 05C38 Paths and cycles

Keywords:

dominating set; star; cubic graph; paths
Full Text:

References:

 [1] Payan, Cahiers C.E.R.O. 17 pp 307– (1975) [2] DOI: 10.1002/jgt.3190130610 · Zbl 0708.05058 [3] DOI: 10.1016/0012-365X(75)90058-8 · Zbl 0323.05127 [4] Arnautov, Prikl. Mat. i Programmirovanie Vyp. 11 pp 3– (1974) [5] DOI: 10.1002/net.3230070305 · Zbl 0384.05051 [6] Blank, Prikl. Math, i Programmirovanie Vyp. 10 pp 3– (1973) [7] Edmonds, Can. J. Math. 17 pp 449– (1964) · Zbl 0132.20903
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