Planar cycle-extendable graphs
Aditya Y Dalwadi, Kapil R Shenvi Pause, Ajit A Diwan, Nishad Kothari

TL;DR
This paper characterizes all planar cycle-extendable graphs, which are graphs where even cycles extend to perfect matchings, using $K_2$ and four infinite families, advancing understanding of their structure.
Contribution
It provides a complete characterization of planar cycle-extendable graphs in terms of basic building blocks and infinite families, filling a gap in the existing literature.
Findings
Characterization of all planar cycle-extendable graphs.
Identification of $K_2$ and four infinite families as fundamental components.
Extension of previous results on claw-free and bipartite planar graphs.
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 Lov\'asz and Plummer. A cycle of a graph is conformal if 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 is cycle-extendable if - for each even cycle - the cycle is conformal, or equivalently, each perfect matching of extends to a perfect matching of , or equivalently, is the symmetric difference of two perfect matchings of…
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Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · graph theory and CDMA systems
