2022/10/14 by Gregory Gutin, Gutin, Gregory, Anders Yeo +1
Computer Science · Materials Science · #Advanced Graph Neural Networks #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Nanocluster Synthesis and Applications
paper · pdf · doi:10.48550/arxiv.2210.07722
openalex publication_date 2022/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 serious 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.