On 1-factors of point determining graphs

✍ Scribed by Dennis P. Geoffroy

• Publisher: Elsevier Science
• Year: 1978
• Tongue: English
• Weight: 532 KB
• Volume: 24
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(78)90196-6

✦ Synopsis

A graph G is said to be point determinie, g if and only if distinct poiuts of G have distinc~ neigh~orhcods~ Fo: such a graph G. the nucleus is defined to !'.e the set G ~ consisting of a!! points v o[ G for ~vhich G-r is a point determini~'g graph.

h~ [4]. Samner exhibited several famiiies of gratq,s H such ~hat if G °:= h r, for some poin! determi~i~ ~ graph G, then G has a l-factor. In this paper, we :x~end this class of graphs, Throughom this paper, atl ~aphs considered will be finite, ~mdirected and without t ?OF, o or multiple edges.

For points a and b of a graph G, we shall write a k b if a and b are adjacent, and am k otherwise. When no co.~fusion is possible, we shah denote the set of points of a graph G by G as well. The neighborhood of a, N(a), is tt-;e set of points of G adjacent to a and the degree of a, deg c:, is the cardinaiity of N(a).

G-a denotes the graph induced by the poims in G -{a}.

Let G be a graph and S a subset of G. S is called a ,Tr-s,:'t of G if and only if N(a) = N(t,) for every a, b ,.-~ S and 5~ !:.s maximal with respect to this property. The set {a, b} then is referred to as a ~r-pair of G.