A characterization of consistent marked graphs

Abstract

A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterizing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis.

Groups of balanced labelings on graphs

“…As in Lemma 6.1 we find ttrails p 1 and p 2 from v j to v i such that s r , p 1 Thus the elements of B ′ (V , A) are, by abuse of notation, such elements of W (V  E, A), which take 0 value of every edge of G. For the case of A = Z 2 this would be identical to describing the consistent marked (vertex-signed) graphs, which were studied and characterized in [1,2,8,11,13,17]. In [12] some relations between consistent marked graphs and balanced signed graphs are studied.…”

Characterization of Signed Graphs Whose Iterated Signed Line Graphs are Balanced or S-Consistent

“…Further, S is said to be consistent if every cycle in it is consistent (Beineke and Harary [10]). Beineke and Harary [10,11] were the first to pose the problem of characterizing consistent marked graphs, which was subsequently settled by Acharya [2,3], Rao [23] and Hoede [19]. Recently, new characterizations of consistent marked graphs have been obtained by Roberts and Xu [25].…”

Canonical Consistency of Semi-Total Line Signed Graphs

“…Theorem 1 [16] A marked graph G l is consistent if and only if for any spanning tree T of G all fundamental cycles with respect to T are consistent and all common paths of pairs of those fundamental cycles have end vertices carrying the same marks.…”

“…Canonically consistent semi-total line sigraphs Beineke and Harary [11] and [12] were the first to pose the problem of characterizing consistent marked graphs, which was subsequently settled by Acharya [13] and [14], Rao [15], Hoede [16] and Roberts & Xu [17]. Acharya and Sinha obtained consistency of sigraphs that satisfy certain sigraph equations in [18] and [19].…”