On the parameterized complexity of non-hereditary relaxations of clique - Archive ouverte HAL Access content directly
Reports (Research Report) Year : 2023

On the parameterized complexity of non-hereditary relaxations of clique

Abstract

We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of Clique: s-Club and s-Clique, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and γ-Complete Subgraph in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that s-Club and s-Clique are NP-hard even restricted to graphs of degeneracy ≤ 3 whenever s ≥ 3, and to graphs of degeneracy ≤ 2 whenever s ≥ 5, which is a strictly stronger result than its W[1]-hardness parameterized by the degeneracy. We also obtain that these problems are solvable in polynomial time on graphs of degeneracy 1. Concerning γ-Complete Subgraph, we prove that it is W[1]-hard parameterized by both the degeneracy, which implies the W[1]-hardness parameterized by the number of vertices in the γ-complete-subgraph, and the number of elements outside the γ-complete subgraph.
Fichier principal
Vignette du fichier
On the parameterized complexity of non-hereditary relaxations of clique.pdf (593.39 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-04036849 , version 1 (26-09-2023)

Licence

Attribution - NonCommercial - ShareAlike

Identifiers

Cite

Ambroise Baril, Antoine Castillon, Nacim Oijid. On the parameterized complexity of non-hereditary relaxations of clique. Liris; Loria; CRIStAL. 2023. ⟨hal-04036849⟩
73 View
24 Download

Altmetric

Share

Gmail Facebook X LinkedIn More