The Complexity of Contracting Bipartite Graphs into Small Cycles

TL;DR

The paper proves that $C_5$-Contractibility and $C_4$-Contractibility are NP-hard on bipartite graphs by constructing two layered reductions from Positive NAE-SAT: first building a non-bipartite gadget graph $H$ and then a bipartite graph $G$ via selective edge subdivisions, with a rigorous equivalence between $H$ and $G$ for the respective targets. It introduces structured witness concepts (C5-witness and a nice C4-witness) and demonstrates how gadget neighborhoods enforce consistent variable assignments, enabling a bidirectional translation between satisfiability and contractibility. The results extend the landscape of cycle contractibility, discuss implications for restricted graph classes (including diameter-2, $K_4$-free graphs), and connect to broader problems like Disconnected Cut. The paper also outlines future directions, such as longer cycles, other restricted families, and potential dichotomies in contractibility problems.

Abstract

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $C_4$-Contractibility is NP-complete in general graphs. It is easy to verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on bipartite graphs for $\ell = 6$ and posed as open problems the status of the problem when $\ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs.

Figures (3)

• Figure 1: (Left) The graph $H$ with certain edges highlighted as purple (dotted) edges denote setting variable $X_i$ to True and as green (dashed) edges denote setting $X_i$ to False, respectively. (Right) The bipartite graph $G$ where blue (round) and red (squares) vertices denote a bipartition.
• Figure 2: (Left) The graph $G$ where only three vertices each in $D$ and $D'$ shown with purple (dotted) edges denote setting variable $X_i$ to True and green (dashed) edges denote setting $X_i$ to False. (Right) Adjacency relation between different subsets of vertices.
• Figure 3: (Left) The graph $G$ with different sets of edges highlighted. Here, purple (dotted) edges denote setting the variable to True and the green (dashed) edges denote setting it to False in a Yes-instance. (Right) Adjacency relation between subset of vertices where $\widehat{P}=S^+ \setminus \{t,t'\}$ and $\widehat{N}=S^- \setminus \{f,f'\}$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Definition 2.1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4