Geodesic
For which graphs does every edge belong to exactly two chordless cycles?
The electronic journal of combinatorics, Tome 3 (1996) no. 1
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A graph is 2-cycled if each edge is contained in exactly two of its chordless cycles. The 2-cycled graphs arise in connection with the study of balanced signing of graphs and matrices. The concept of balance of a $\{0,+1,-1\}$-matrix or a signed bipartite graph has been studied by Truemper and by Conforti et al. The concept of $\alpha$-balance is a generalization introduced by Truemper. Truemper exhibits a family ${\cal F}$ of planar graphs such that a graph $G$ can be signed to be $\alpha$-balanced if and only if each induced subgraph of $G$ in ${\cal F}$ can. We show here that the graphs in ${\cal F}$ are exactly the 2-connected 2-cycled graphs.
DOI : 10.37236/1238
Classification : 05C38, 05C50
Mots-clés : chordless cycles, 2-cycled graphs, matrices, balance 
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     author = {Uri N. Peled and Julin Wu},
     title = {For which graphs does every edge belong to exactly two chordless cycles?},
     journal = {The electronic journal of combinatorics},
     year = {1996},
     volume = {3},
     number = {1},
     doi = {10.37236/1238},
     zbl = {0851.05072},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1238/}
}
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Uri N. Peled; Julin Wu. For which graphs does every edge belong to exactly two chordless cycles?. The electronic journal of combinatorics, Tome 3 (1996) no. 1. doi: 10.37236/1238

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