Vertex identification to a forest

TL;DR

This work initiates the study of vertex identification as a graph modification operation to reach a target property, focusing on the forest class. It formalizes $id_{rak{H}}(G)$ as the minimum size of the identifying set and analyzes the Identification to Forest problem. The authors establish NP-completeness for forests and derive a linear kernel of size $2k+1$ by connecting the problem to Vertex Cover on bridgeless graphs, plus a bridgeless augmentation to maintain equivalence. They further develop obstructions for $ ext{F}^{(k)}$, prove a universal obstruction of idf via a triplet of minor-closed families, and discuss the broader relation to contraction, identification minors, and open questions. The results lay groundwork for parameterized algorithms and structural understanding of vertex identifications in minor-closed classes, with potential applications to kernelization and obstruction theory.

Abstract

Let $\mathcal{H}$ be a graph class and $k\in\mathbb{N}$. We say a graph $G$ admits a \emph{$k$-identification to $\mathcal{H}$} if there is a partition $\mathcal{P}$ of some set $X\subseteq V(G)$ of size at most $k$ such that after identifying each part in $\mathcal{P}$ to a single vertex, the resulting graph belongs to $\mathcal{H}$. The graph parameter ${\sf id}_{\mathcal{H}}$ is defined so that ${\sf id}_{\mathcal{H}}(G)$ is the minimum $k$ such that $G$ admits a $k$-identification to $\mathcal{H}$, and the problem of \textsc{Identification to $\mathcal{H}$} asks, given a graph $G$ and $k\in\mathbb{N}$, whether ${\sf id}_{\mathcal{H}}(G)\le k$. If we set $\mathcal{H}$ to be the class $\mathcal{F}$ of acyclic graphs, we generate the problem \textsc{Identification to Forest}, which we show to be {\sf NP}-complete. We prove that, when parameterized by the size $k$ of the identification set, it admits a kernel of size $2k+1$. For our kernel we reveal a close relation of \textsc{Identification to Forest} with the \textsc{Vertex Cover} problem. We also study the combinatorics of the \textsf{yes}-instances of \textsc{Identification to $\mathcal{H}$}, i.e., the class $\mathcal{H}^{(k)}:=\{G\mid {\sf id}_{\mathcal{H}}(G)\le k\}$, {which we show to be minor-closed for every $k$} when $\mathcal{H}$ is minor-closed. We prove that the minor-obstructions of $\mathcal{F}^{(k)}$ are of size at most $2k+4$. We also prove that every graph $G$ such that ${\sf id}_{\mathcal{F}}(G)$ is sufficiently big contains as a minor either a cycle on $k$ vertices, or $k$ disjoint triangles, or the \emph{$k$-marguerite} graph, that is the graph obtained by $k$ disjoint triangles by identifying one vertex of each of them into the same vertex.

Figures (5)

• Figure 1: The obstructions of $\mathcal{V}_k$ (first and second columns) and $\mathcal{F}^{(k)}$ (second and third columns) for $k\le3$. Each graph in ${\sf obs}(\mathcal{F}^{(k)})$ is either 1) also a graph in ${\sf obs}(\mathcal{V}^{(k)})$ (second column), or 2) can be obtained from a graph in ${\sf obs}(\mathcal{V}^{(k)})$ with bridges (first column) by adding edges (in blue in the third column), or 3) is also a graph in ${\sf obs}(\mathcal{V}^{(k+1)})$ (in purple in the third column). We use yellow shadows for disconnected obstructions, to make clear that each of them is a single graph.
• Figure 2: Graphs $G$ and $G/uv$.
• Figure 3: Graphs $F$, $F_x$, and $F_x'.$
• Figure 4: The universal obstruction for Identification to Forest.
• Figure 5: The graph $H_k$ for $k=5$. We give credit to Hugo Jacob for finding it.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• proof
• proof
• Lemma 2
• proof
• Lemma 3