
TL;DR
This paper characterizes when a graph contains a rooted $K_4$-minor at four specified vertices, identifying six classes of obstructions and providing conditions for 4-connected non-planar and 3-connected planar graphs.
Contribution
It provides a precise characterization of rooted $K_4$-minors through six classes of obstructions, extending understanding of graph minors with specific root vertices.
Findings
Six classes of obstructions identified
4-connected non-planar graphs always contain the rooted $K_4$-minor
3-connected planar graphs contain the rooted $K_4$-minor iff roots are not on a single face
Abstract
Let be four vertices in a graph . A \emph{-minor rooted} at consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of , respectively containing . We characterise precisely when contains a -minor rooted at by describing six classes of obstructions, which are the edge-maximal graphs containing no -minor rooted at . The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a -minor rooted at for every choice of . (2) A 3-connected planar graph contains a -minor rooted at if and only if are not on a single face.
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