Forbidding the subdivided claw as a subgraph or a minor

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 (finite and infinite) 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. We also characterize graphs that forbid each subtree of $Y$. We discuss the relevance of our results for Turán. numbers of trees, and pathwidth and growth constants for graphs without a particular tree as a minor.

💡 Research Summary

The paper studies the graph Y, the “subdivided claw”: a 7‑vertex tree obtained by subdividing each edge of the claw K₁,₃ exactly once. Because every vertex of Y has degree at most 3, forbidding Y as a subgraph, as a topological minor, or as a (usual) minor are equivalent conditions. The motivation comes from vertex‑contraction‑deletion (VCD) minors: a VCD‑minor is obtained by repeatedly deleting vertices or contracting all edges incident with a single vertex. In a line graph L(H) with matching number at least 4, forbidding a K₁,₃‑VCD‑minor is equivalent to forbidding Y as a subgraph of H. Hence a structural description of Y‑free graphs is a key step toward characterising K₁,₃‑VCD‑minor‑free line graphs.

The authors introduce a family of building blocks called beads. A bead is one of four types: * K₄ (with one designated primary vertex), * K₂,₁,₁ (the complete bipartite graph with parts of size 2, 1, 1, primary vertices are the two‑vertex part), * K₁,₁, t₁ (t₁ ≥ 0) – a star with two leaves designated primary, * K₂, t₂ (t₂ ≥ 2) – a complete bipartite graph with a part of size 2 (the primary vertices) and a part of size t₂.

Beads may be strung together by identifying a primary vertex of one bead with a primary vertex of another; each primary vertex may be identified with at most one other primary vertex. The auxiliary bipartite graph A records beads (black vertices) and primary vertices (white vertices). If A is a path, the resulting graph is a strand; if A is a cycle, it is a necklace. A spiked strand (or spiked necklace) is a strand (necklace) that may have additional pendant edges (spikes) attached to primary vertices that belong to exactly two beads. All such graphs form the family 𝔅; they are connected by construction.

Another operation is cloning leaves. Two non‑adjacent vertices u, v are clones if N(u)=N(v). A clone class consisting of leaves may be enlarged arbitrarily by adding new vertices with the same neighbourhood; this is called leaf cloning.

The main theorem (Theorem 1.2) states:

A connected graph G contains no subgraph isomorphic to Y (equivalently, is Y‑minor‑free) iff either
(a) G can be obtained from a graph on at most six vertices by optionally cloning leaves, or
(b) G belongs to the family 𝔅 (i.e., G is a spiked strand or a spiked necklace).

The proof proceeds by selecting a longest path P=v₀v₁…v_ℓ in G and analysing the set L_i = N(v_i) \ V(P) of vertices outside the path that are adjacent to v_i. Lemma 2.1 shows that P is edge‑dominating, so all vertices outside P lie in the independent set L = ⋃L_i. A series of lemmas (2.2–2.13) establish strict constraints on how the L_i’s may intersect, on the existence of chords (edges v_i v_j with |i−j|≥2), on “vees” (paths v_i w v_{i+2} with w∉P), and on 2‑chords. The key observations are:

• Adjacent L_i’s are disjoint (Lemma 2.3).
• If L_i and L_j intersect for i<j, then j=i+2 or (i,j)=(1,ℓ−1) (Lemma 2.4).
• No vertex outside P can belong to three different L_i’s (Lemma 2.5).
• A vee cannot cross a 2‑chord or another vee (Lemma 2.6).
• The presence of a chord v₀v_i forces emptiness of certain nearby L‑sets (Lemma 2.7).

When ℓ≤4, the graph is either tiny (≤6 vertices) or a spiked strand, as shown in Lemmas 2.8–2.10. For ℓ≥5, the above constraints force the structure of G to be a concatenation of beads, possibly with spikes, i.e., a member of 𝔅. Conversely, any graph of type (a) or (b) clearly avoids Y, because each bead is Y‑free and spikes are pendant edges that cannot create the required three‑branch configuration.

The authors then extend the classification to every subtree T of Y. For each such T they give an analogous description: the allowed beads are those that do not contain T, and the resulting graphs are again spiked strands or necklaces with restricted bead types. Infinite analogues are treated in Section 7, showing that the same structural description holds for infinite connected Y‑free graphs, with the only difference being that the auxiliary graph A may be an infinite path or cycle.

In Section 8 the paper discusses broader implications. Since Y‑free graphs have bounded pathwidth (the bead construction yields a tree‑decomposition of width at most 2), they also have linear‑time recognisability. The authors note that Turán‑type extremal results for trees often hinge on the maximum number of edges in a Y‑free graph; their structural theorem implies that the extremal graphs are essentially dense beads (cliques) with many cloned leaves. Moreover, the growth constant (the exponential rate of the number of labelled Y‑free graphs) is determined by the generating function of the bead family, showing that forbidding a small tree can dramatically restrict the asymptotic enumeration.

Overall, the paper delivers a complete and elegant characterisation of graphs that forbid the subdivided claw Y, both in the finite and infinite settings, and connects this classification to several classical topics in extremal and structural graph theory.