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By J. Akiyama, Y. Egawa and H. Enomoto (Eds.)

ISBN-10: 0444705384

ISBN-13: 9780444705389

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Sample text

One can observe from Corollary 4c, Corollary 2g, and Theorem 3 that there are four equivalent definitions of the domination number of a graph G = (V, E): (i) the minimum number of vertices in a set D such that every vertex in V - D is adjacent to at least one vertex in D ; Gallai theorems 41 (ii) the minimum number of vertices in a transversal of the closed neighborhoods of G; (iii) the minimum order of a spanning star partition of the vertices of G (note; this does not allow isolated vertices); (iv) the minimum number of trees in a spanning forest in which every tree has diameter G2 (note: this allows isolated vertices).

Let S = V ( G ) and let a set X = { x , , x 2 , . . e. there exists a set of t independent edges ( x i , f ( x i ) ) , i = 1,2, . . , t. Let aL((uJ denote the maximum (minimum) number of vertices in a minimal transversal of non-matchable sets, and let B'(,S-) denote the maximum (minimum) number of vertices in a maximal matchable set. Notice that B + ( P ) = p,(G) (the matching number). Corotlary 2e. For any nontrivial, connected graph G with p vertices, ( 9 a, + P1 = P (ii) a; + fl- = p . From Corollary 2e(i) and Gallai's Theorem, Part 11, we may immediately conclude Corollary 2f.

Moreover, by an even cycle partition ( E C P ) of r, we mean a set = { C , , . . , c k } of vertex-disjoint cycles of rwhose union is V ( T ) . Hence t? is a partition of V ( T ) into even cycles. It is easy to see that each (PMP) P = [PI,P,, P3] gives rise to exactly three (ECP)’s, namely, = PI U P,, C, = P, U P3 and = P3 U PI. On the other hand, for each (ECP) = {Cl, . . , because each even cycle has two perfect matchings and the set of all edges not in C1,. . , C, also form a perfect matching of r.

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Proceedings of the First Japan Conference on Graph Theory and Applications, Hakone, Japan, June 1-5, 1986 by J. Akiyama, Y. Egawa and H. Enomoto (Eds.)


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