(1,1)-Cluster Editing is polynomial-time solvable - Gutin, Gregory; Yeo, Anders (Discrete Applied Mathematics, 2023)
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)→{0,1,…,a} and d∗:V(G)→{0,1,…,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∈V(G) and adding at most a∗(v) edges incident to every v∈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 series of five polynomial-time reductions to C3-free and C4-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving (1,1)-Cluster Editing on C3-free and C4-free graphs of maximum degree at most 3.