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Abstract for Uri N. Peled and Julin Wu, For Which Graphs Does Every Edge Belong to Exactly Two Chordless Cycles?

A graph is {\it 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 {\it 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.

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