The Antisymmetric Line Graph

Abstract

We introduce the \emph{antisymmetric line graph} $\mathcal{A}(G)$, a canonical signed refinement of the line graph defined on the oriented edges of a graph $G$ via an antisymmetric incidence rule. We show that $\mathcal{A}(G)$ is balanced if and only if $G$ is bipartite, so the frustration index $\ell(\mathcal{A}(G))$ defines a switching-invariant measure of non-bipartiteness. The switching class of $\mathcal{A}(G)$ determines $G$ up to isomorphism modulo isolated vertices, thereby resolving Whitney's exceptional ambiguity. We obtain quantitative bounds relating $\ell(\mathcal{A}(G))$ to classical measures of non-bipartiteness; in particular, \[ \operatorname{def}(G)\le \ell(\mathcal{A}(G))\le (Δ(G)-1)\operatorname{def}(G), \] where $\operatorname{def}(G)$ is the minimum number of edges whose deletion makes $G$ bipartite. Our strongest result is an exact identification in the cubic case: for every cubic graph $G$, with $\operatorname{oct}(G)$ denoting the odd cycle transversal number, \[ \ell(\mathcal{A}(G)) = 2\operatorname{oct}(G). \] Hence on cubic graphs a canonical signed line-graph invariant recovers a central bipartization parameter, and computing $\ell(\mathcal{A}(G))$ is NP-hard even for cubic inputs.

Theorems & Definitions (43)

• Theorem 1: Whitney Whitney1932
• Definition 1: Switching class
• Lemma 1: ALG refines the line graph
• proof
• Definition 2: Antisymmetric Line Graph
• Remark 1: Laplacian spectrum versus switching class
• Proposition 1: Switching invariance
• proof
• Proposition 2: Canonical signing
• proof