Planar cycle-extendable graphs

TL;DR

The paper resolves the planar cycle-extendability problem for irreducible matching-covered graphs by combining tight-cut decomposition, ear-decomposition theory, and Norine-Thomas brick structure. It introduces the osculating bicycle and half-biwheel constructions to organize planar irreducible graphs into four infinite families (\mathcal{G}_0–\mathcal{G}_3) plus the base case $K_2$, and proves that every planar irreducible cycle-extendable graph belongs to this catalog. The work leverages bricks/braces decompositions, the Unique Tight Cut Decomposition Theorem, and the Norine-Thomas generation framework to obtain a complete, constructive characterization and to place the cycle-extendability decision problem for planar graphs in NP ∩ P. These results yield practical classifications for planar graphs and provide new structural insights into how alternating cycles and ear decompositions interact under planarity constraints.

Abstract

For most problems pertaining to perfect matchings, one may restrict attention to matching covered graphs - that is, connected nontrivial graphs with the property that each edge belongs to some perfect matching. There is extensive literature on these graphs that are also known as 1-extendable graphs (since each edge extends to a perfect matching) including an ear decomposition theorem due to Lovász and Plummer. A cycle $C$ of a graph $G$ is conformal if $G-V(C)$ has a perfect matching; such cycles play an important role in the study of perfect matchings, especially when investigating the Pfaffian orientation problem. A matching covered graph $G$ is cycle-extendable if - for each even cycle $C$ - the cycle $C$ is conformal, or equivalently, each perfect matching of $C$ extends to a perfect matching of $G$, or equivalently, $C$ is the symmetric difference of two perfect matchings of $G$, or equivalently, $C$ extends to an ear decomposition of $G$. In the literature, these are also known as cycle-nice or as 1-cycle resonant graphs. Zhang, Wang, Yuan, Ng and Cheng, 2022, provided a characterization of claw-free cycle-extendable graphs. Guo and Zhang, 2004, and independently Zhang and Li, 2012, provided characterizations of bipartite planar cycle-extendable graphs. In this paper, we establish a characterization of all planar cycle-extendable graphs - in terms of $K_2$ and four infinite families.

Figures (21)

• Figure 1: (a) the Cube graph is not cycle-extendable; (b) a bipartite cycle-extendable graph; (c) a nonbipartite cycle-extendable graph $W_5^{-}$
• Figure 2: a bipartite ear decomposition of a bipartite matching covered graph
• Figure 3: an ear decomposition of $R_8^{-}$
• Figure 4: a matching covered graph and its nontrivial tight cuts
• Figure 5: illustration for the proof of Theorem \ref{['DegreeThree-Graph']}

Theorems & Definitions (17)

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