(1,1)-Cluster Editing is polynomial-time solvable
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 clust...
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| Vydáno v: | Discrete Applied Mathematics Ročník 340; s. 259 - 271 |
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| Jazyk: | angličtina |
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Elsevier B.V
15.12.2023
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| ISSN: | 0166-218X, 1872-6771 |
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| 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)→{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. |
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| AbstractList | 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. |
| Author | Gutin, Gregory Yeo, Anders |
| Author_xml | – sequence: 1 givenname: Gregory orcidid: 0000-0002-2377-0417 surname: Gutin fullname: Gutin, Gregory email: g.gutin@rhul.ac.uk organization: Department of Computer Science. Royal Holloway University of London, United Kingdom – sequence: 2 givenname: Anders surname: Yeo fullname: Yeo, Anders email: yeo@imada.sdu.dk organization: Department of Mathematics and Computer Science, University of Southern Denmark, Denmark and Department of Pure and Applied Mathematics, University of Johannesburg, South Africa |
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| Keywords | Cluster editing Polynomial algorithm |
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| Snippet | 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... |
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| Title | (1,1)-Cluster Editing is polynomial-time solvable |
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