(1,1)-Cluster Editing is Polynomial-time Solvable

TL;DR

This paper proves that the (1,1)-Cluster Editing problem, which involves transforming a graph into a cluster graph with limited edge modifications per vertex, can be solved in polynomial time, confirming a longstanding conjecture.

Contribution

The authors resolve Abu-Khzam's conjecture by developing polynomial-time reductions and an algorithm for (1,1)-Cluster Editing on specific graph classes.

Findings

(1,1)-Cluster Editing is polynomial-time solvable.

Provided five polynomial-time reductions to special graph classes.

Designed an efficient algorithm for the reduced problem.

Abstract

A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$ and $d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\in V(G)$. Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3.