On the complexity of edge subdivision to $H$-free graphs

Abstract: Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$ in $G$ by at most $k$ edge subdivisions. We show that the problem is polynomial-time solvable when every component of $H$ is a subdivided star or a subdivided bistar, and at most one component is a subdivided bistar. On the other hand, we prove that \textsc{$H$-free Subdivision} is NP-complete and, assuming the Exponential Time Hypothesis, admits no $2^{o(k)} n<sup>{O(1)}$-time algorithm whenever $H$ satisfies any of the following conditions: \begin{itemize} \item $H$ has minimum degree at least $2$, and the neighborhood of every degree-$2$ vertex induces a $K_2$; \item the vertices of degree at least $3$ in $H$ induce a graph with at least two edges; \item $H$ has a triangle with two vertices of degree at least $3$; \item $H$ contains, as an induced subgraph, the graph obtained from two vertex-disjoint triangles by adding one edge between them; \item $H$ contains exactly one triangle; \item $H$ has girth at least $4$; \item $H$ is a tree with exactly two vertices of degree at least $3$ at distance $2$ or at least $4$. \end{itemize} A simple bounded search-tree algorithm for the problem runs in $2^{O(k)} n<sup>{O(1)}$ time. Thus, for all hardness cases above, this running time is essentially optimal under ETH.

• The paper introduces H-free Subdivision and gives a 2^{O(k)}·n^{O(1)} fixed-parameter algorithm by branching on edges in induced copies of H.
• The paper proves NP-completeness and rules out 2^{o(k)}·n^{O(1)} algorithms under ETH for broad families, including non-forest graphs of girth at least 4, graphs with one triangle, and many trees with two branching vertices.
• The paper identifies polynomial-time cases involving subdivided stars and bistars while leaving a narrow gap, including trees whose two branching vertices are at distance 3 and some graphs with multiple bistar components.

Problem definition and context

The paper studies a new graph modification problem in which the permitted operation is edge subdivision: replacing an edge $uv$ by a path $u$–$w$–$v$ through a fresh vertex $w$. For a fixed graph $H$, the $H$-free Subdivision problem takes as input a graph $G$ and an integer $k$ and asks whether $G$ can be made free of induced copies of $u$0 by at most $u$1 successive edge subdivisions. The authors position this problem within the established landscape of modification problems targeting hereditary classes of the form "no induced copy of $u$2", where the operation ranges over vertex deletion, edge deletion/completion/editing, contraction, subgraph complementation, switching, and vertex splitting; for most of these operations, NP-completeness is known for essentially all nontrivial choices of $u$3.

Two structural observations frame the entire paper. First, subdivisions never destroy minors or induced minors, so the problem is trivial for minor-closed and induced-minor-closed targets. Second, within the hereditary setting, an induced subdivided star (a tree with at most one vertex of degree at least 3) can never be destroyed by subdivision, and the only way to destroy an induced subdivided bistar (a tree whose two degree-3 vertices are adjacent) is to subdivide the edge joining them. This yields a polynomial-time solvable case: if every component of $u$4 is a subdivided star or subdivided bistar, with at most one bistar component, the problem is trivially decidable. The authors conjecture that these are the only polynomial-time cases, and that all remaining cases are NP-complete with no $u$5 algorithm under the Exponential Time Hypothesis (ETH). The paper substantiates this conjecture for a broad family of graphs $u$6.

Upper bound

A bounded search tree yields a straightforward FPT algorithm running in time $u$7: repeatedly find an induced copy of $u$8 and branch on which of its edges to subdivide. All hardness results below are tight against this bound under ETH, since they rule out $u$9 algorithms.

Hardness via equivalence with edge deletion

The first hardness theorem covers two conditions on $w$0: (i) $w$1 and the neighborhood of every degree-2 vertex induces a $w$2; or (ii) the vertices of degree at least 3 induce at least two edges. Case (i) is handled by showing polynomial equivalence between $w$3-free Subdivision and $w$4-free Edge Deletion. The key insight is that a subdivision vertex has degree 2 with a non-adjacent neighborhood, so it cannot play the role of any vertex of such an $w$5; hence subdividing exactly the edges of an edge-deletion solution produces an $w$6-free graph, and conversely any subdivision solution induces an edge-deletion solution on the original edges. Since $w$7-free Edge Deletion is NP-complete and admits no $w$8 algorithm whenever $w$9, hardness transfers directly.

Case (ii) uses a reduction from $v$0-free Edge Deletion where $v$1 is the subgraph induced by the high-degree vertices. The construction attaches, for each component $v$2 of $v$3 (where $v$4 is the high-degree set), either $v$5 disjoint copies, per-vertex copies, or per-pair copies of $v$6 to the input graph, so that any attempted induced copy of $v$7 can be completed to a copy of $v$8 using an untouched gadget. The parameter is preserved, giving the ETH lower bound.

Girth-3 graphs

For graphs $v$9 of girth 3 not covered above, triangles have a restricted form: a hanging triangle contains exactly one high-degree vertex, and a roof triangle contains exactly two. The paper proves NP-completeness and the matching ETH lower bound when $w$0 has a roof triangle, or when it has two hanging triangles whose attachment vertices are adjacent. Both reductions start from an appropriate $w$1-free Edge Deletion instance, where $w$2 is the book-like subgraph spanned by the roof-triangle vertices (or by the adjacent attachment vertices together with their hanging triangles). For every edge of the deletion instance, the construction attaches $w$3 copies of the remainder $w$4 — in both orientations when the roles of $w$5 and $w$6 are asymmetric — guaranteeing that at least one attached copy survives any solution of size at most $w$7. Subdivisions cannot create triangles, so any induced copy of $w$8 must realize its triangle part inside the original graph, forcing an induced copy of $w$9 to survive unless the corresponding edges are deleted. The paper also proves hardness for every $H$0 containing precisely one triangle; this follows from the later girth-at-least-4 machinery, since the shortest-cycle vertices then induce a triangle, which is 2-connected.

Non-forest graphs of girth at least 4

The main technical contribution is a complete hardness proof for every non-forest $H$1 with $H$2. The generic reduction is from Vertex Cover. Given $H$3, the construction places $H$4 as an independent set, adds a universal vertex $H$5, and for each edge $H$6 introduces a private induced copy of $H$7 in which $H$8 plays the role of a degree-2 vertex $H$9 on a shortest cycle and $H$0 play the roles of its neighbors. Crucially, all pairs of distinct edge-gadgets are made complete to each other, and every non-endpoint vertex of $H$1 is made complete to every gadget. Any induced copy of $H$2 in the resulting graph is triangle-free, so it can meet at most two gadgets; a careful case analysis shows that any copy distributed across two gadgets would force $H$3 to be an induced subgraph of one of three explicitly defined exceptional classes $H$4. If $H$5 lies outside these classes, a vertex cover translates into subdividing the edges $H$6 for $H$7 in the cover, and the counting argument on shortest cycles shows this destroys all copies of $H$8 without creating new ones.

For $H$9 inside the exceptional classes, a second reduction applies, again from Vertex Cover but on instances of girth exceeding $G$0, so that short cycles are confined to individual gadgets. Two situations arise: either the vertices lying on shortest cycles induce a 2-connected subgraph (so those cycles cannot be split across gadgets intersecting in one vertex), or the core of $G$1 is one of a finite family of explicit structures — the 8-vertex graph $G$2, or the graphs $G$3 consisting of two bundles of internally disjoint 2-paths between $G$4 and $G$5 with optional adjacency or subdivided adjacency to a vertex $G$6. In the latter case the construction splits $G$7 at the cut vertex $G$8, attaching one part to ordered edges and the other to endpoints, so that every copy of $G$9 still corresponds to an edge of the source graph. A final structural lemma shows these reductions plus the earlier results cover all non-forest graphs of girth at least 4, completing the dichotomy for this family.

Trees

For trees $k$0 with exactly two branching vertices $k$1, the paper proves hardness when $k$2 is even (including distance 2), or odd and at least 5. Even distances reduce from $k$3-free Edge Deletion; odd distances at least 5 reduce from $k$4-free Edge Deletion. Both reductions attach, to every vertex of the input graph, $k$5 copies of rooted trees $k$6 for each offset $k$7 up to roughly half the branching distance $k$8, where $k$9 scales the budget ($G$0). An induced $G$1 or $G$2 surviving in the pruned input graph can always be extended, via untouched gadget copies, into an induced copy of $G$3; conversely, subdividing each deleted edge $G$4 times eliminates all copies because the branching vertices' connecting path would need to traverse an induced path too long to exist. The counting argument relies on having strictly more gadget copies than the budget allows the solution to touch.

Limitations and open questions

The central conjecture remains open in specific cases. Among trees with exactly two branching vertices, the case where the branching vertices are at distance 3 is not covered: the authors state plainly that they lack a reduction yielding the parameterized subexponential lower bound there, although the general $G$5 upper bound still applies. More broadly, the full conjecture — that NP-completeness with the ETH lower bound holds for every $G$6 outside the polynomial-time family, including graphs with multiple bistar components — is verified only for the families listed above. The reductions also inherit their lower bounds from $G$7-free Edge Deletion and high-girth Vertex Cover, so any improvement in those base bounds would transfer, but no stronger unconditional lower bounds are claimed.

Conclusion

This paper initiates the study of $G$8-free Subdivision and establishes a near-dichotomy: polynomial-time solvability when every component of $G$9 is a subdivided star or bistar (with at most one bistar), and NP-completeness with a tight $u$00 ETH lower bound — matching the trivial $u$01 search-tree algorithm — for a wide range of $u$02, including all non-forest graphs of girth at least 4, all graphs with exactly one triangle, several structured girth-3 families, and trees with two branching vertices at distance 2 or at least 4. The remaining gap between the proven-hard and provably-easy cases is narrow and explicitly identified, making the resolution of the stated conjecture a well-defined target for subsequent work.