A characterization of consistent marked graphs

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 characterzing 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

Characterization of Line-Consistent Signed Graphs

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede's relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede's theorem as well as a structural description of line-consistent signed graphs.Comment: 5 pages. V2 defines sign of a walk and corrects statement of Theorem 4 ("is balanced and" was missing); also minor copyeditin

The H-Line Signed Graph of a Signed Graph

For standard terminology and notion in graph theory we refer the reader to Harary; the non-standard will be given in this paper as and when required. We treat only finite simple graphs without self loops and isolates