Forbidding the subdivided claw as a subgraph or a mino

TL;DR

The paper addresses the problem of characterizing graphs that forbid the subdivided claw $Y$ as a subgraph or minor, revealing an equivalence between $Y$-minor-free and $Y$-subgraph-free graphs. It develops a bead-based structural framework and distinguishes two principal regimes based on the presence of long cycles: spiked strands (no long cycle) and spiked necklaces (long cycle). The main contribution is a precise dichotomy: every connected $Y$-free graph is either a leaf-clone of a graph on at most six vertices or belongs to the bead family $\\mathcal{B}$, with implications for $K_{1,3}$-VCD-minor-free line graphs. The work also yields pathwidth bounds and raises broader questions about the structure and growth constants of tree-minor-free graph classes, guiding future research in minor-closed graph theory.

Abstract

Let $Y$ be the subdivided claw, the $7$-vertex tree obtained from a claw $K_{1,3}$ by subdividing each edge exactly once. We characterize the graphs that do not have $Y$ as a subgraph, or, equivalently, do not have $Y$ as a minor. This work was motivated by a problem involving VCD minors. A graph $H$ is a vertex contraction-deletion minor, or VCD minor, of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions or contractions of all edges incident with a single vertex. Our result is a key step in describing $K_{1,3}$-VCD-minor-free line graphs. Moreover, our characterization raises some general questions about the structure of graphs forbidding a specific tree as a minor.

Figures (2)

• Figure 1.1: Beads
• Figure 1.2: Structures in $\mathcal{B}$

Theorems & Definitions (69)

• Theorem 1.1: West west; see also kl-fmpw2
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof