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The Minimum Subgraph Complementation Problem

Juan Gutiérrez, Sagartanu Pal

TL;DR

This work studies Minimum Subgraph Complementation (MSC), the optimization variant of Subgraph Complementation, and presents polynomial-time algorithms for several natural graph classes and input types. The authors provide structural characterizations and transfer principles (e.g., complement-transfer) to derive tractable MSC cases across bipartite, split, and co-bipartite graphs, including cases where the input is biregular or a forest and targets involve chordal graphs, fixed degeneracy, or connectivity changes. Key results include poly-time MSC for transforming between bipartite, co-bipartite, and split graphs; a vertex-cover/2K2-free characterization for bipartite-to-chordal completions; an efficient algorithm for forests to fixed degeneracy; a BC-tree-based method to reach 2-connected targets; and a split-decomposition approach to achieve disconnection. Collectively, these findings advance understanding of how far a graph is from a target property under subgraph complementation and yield practical algorithms for several graph-editing settings.

Abstract

Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph \(G\) and a target class \(\mathcal{C}\), the Minimum Subgraph Complementation problem asks for a minimum-size vertex set \(S\) such that complementing the subgraph induced by \(S\) transforms \(G\) into a graph belonging to \(\mathcal{C}\). While the decision version of Subgraph Complementation has been extensively studied and is NP-complete for many graph classes, the algorithmic complexity of its optimization variant has remained largely unexplored. In this paper, we study MSC from an algorithmic perspective. We present polynomial-time algorithms for MSC in several nontrivial settings. Our results include polynomial-time solvability for transforming graphs between bipartite, co-bipartite, and split graphs, as well as for complementing bipartite regular graphs into chordal graphs. We also show that MSC to the class of graphs of fixed degeneracy can be solved in polynomial time when the input graph is a forest. Moreover, we investigate MSC with respect to connectivity and prove that MSC to the class of disconnected graphs and to the class of 2-connected graphs can be solved in polynomial time for arbitrary inputs.

The Minimum Subgraph Complementation Problem

TL;DR

This work studies Minimum Subgraph Complementation (MSC), the optimization variant of Subgraph Complementation, and presents polynomial-time algorithms for several natural graph classes and input types. The authors provide structural characterizations and transfer principles (e.g., complement-transfer) to derive tractable MSC cases across bipartite, split, and co-bipartite graphs, including cases where the input is biregular or a forest and targets involve chordal graphs, fixed degeneracy, or connectivity changes. Key results include poly-time MSC for transforming between bipartite, co-bipartite, and split graphs; a vertex-cover/2K2-free characterization for bipartite-to-chordal completions; an efficient algorithm for forests to fixed degeneracy; a BC-tree-based method to reach 2-connected targets; and a split-decomposition approach to achieve disconnection. Collectively, these findings advance understanding of how far a graph is from a target property under subgraph complementation and yield practical algorithms for several graph-editing settings.

Abstract

Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph and a target class , the Minimum Subgraph Complementation problem asks for a minimum-size vertex set such that complementing the subgraph induced by transforms into a graph belonging to . While the decision version of Subgraph Complementation has been extensively studied and is NP-complete for many graph classes, the algorithmic complexity of its optimization variant has remained largely unexplored. In this paper, we study MSC from an algorithmic perspective. We present polynomial-time algorithms for MSC in several nontrivial settings. Our results include polynomial-time solvability for transforming graphs between bipartite, co-bipartite, and split graphs, as well as for complementing bipartite regular graphs into chordal graphs. We also show that MSC to the class of graphs of fixed degeneracy can be solved in polynomial time when the input graph is a forest. Moreover, we investigate MSC with respect to connectivity and prove that MSC to the class of disconnected graphs and to the class of 2-connected graphs can be solved in polynomial time for arbitrary inputs.
Paper Structure (6 sections, 30 theorems, 19 equations, 2 figures)

This paper contains 6 sections, 30 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $(k,\ell)$-graphs
  3. Bipartite graphs to chordal graphs
  4. Forests to graphs of degeneracy $k$
  5. Increasing the connectivity of a graph
  6. Disconnecting a graph

Key Result

Theorem 1

Let $\mathcal{C}$ be the class of co-bipartite graphs. Then MSC to $\mathcal{C}$ can be solved in polynomial time when the input graph is bipartite.

Figures (2)

  • Figure 1: Situations in the proof of Lemma. \ref{['lemma:bipartite-split-characterization-1']}. $(a)$$S \cap I = \emptyset$. $(b)$$S \cap I \neq \emptyset$. A dashed line implies the absence of an edge in the original graph.
  • Figure 2: $(a)$ A split $(A_2,A_1,B_1,B_2)$ of a graph $G$. $(b)$ The graph $G_A$.

Theorems & Definitions (70)

  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Claim 3
  • proof : Proof of Claim \ref{['claim:complement-commutes']}
  • Corollary 4
  • Theorem 6
  • proof
  • Corollary 7
  • ...and 60 more