On complementarily transitivity of graphs
•We show that the Maximum Complementarily Transitivity Decision Problem is NP-complete for chordal graphs and bipartite graphs. We also show that the decision version of this problem is still NP-complete for two subclasses of bipartite graphs: star-convex bipartite graphs and perfect elimination bip...
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| Vydáno v: | Theoretical computer science Ročník 1058; s. 115582 |
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| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
22.12.2025
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| Témata: | |
| ISSN: | 0304-3975 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | •We show that the Maximum Complementarily Transitivity Decision Problem is NP-complete for chordal graphs and bipartite graphs. We also show that the decision version of this problem is still NP-complete for two subclasses of bipartite graphs: star-convex bipartite graphs and perfect elimination bipartite graphs.•We prove that Maximum Complementarily Transitivity Problem can be solved in linear time in bipartite chain graphs by showing a relation between Trcp(G) and Tr(G). Furthermore, we prove that this problem can be solved in polynomial time for trees by showing that it is equal to TTr(T).•We show three upper bounds for split graphs and give examples of split graphs where the difference between these upper bounds and Trcp(G) can be arbitrarily large.
Let G=(V,E) be a graph where V and E are the vertex and edge sets, respectively. For two disjoint subsets A and B of V, we say AdominatesB if every vertex of B is adjacent to at least one vertex of A in G. A vertex partition π={V1,V2,…,Vk} of G is called a transitive partition of size k if Vi dominates Vj for all 1≤i<j≤k. A subset D of V(G) is called a complementarily dominating set in G if for each vertex x∈V(G)∖D, there exist vertices y,z∈D such that y is adjacent to x and z is not adjacent to x. For two disjoint subsets A and B, we say Acomplementarily dominatesB if, for every vertex x∈B, there exist two vertices y,z∈A, such that y is adjacent to x and z is not adjacent to x. A vertex partition π={V1,V2,…,Vk} of G is called a complementarily transitive partition or aTrcp-partition of size k if Vi complementarily dominates Vj for all 1≤i<j≤k. The maximum integer k for which the above partition exists is called complementarily transitivity of G, and it is denoted by Trcp(G). A complementarily transitive partition of order Trcp(G) is called a Trcp(G)-partition. The Maximum Complementarily Transitivity Problem is to find a complementarily transitive partition of a given graph with the maximum number of parts. In this article, we study this variation of transitive partition from a structural and algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs, perfect elimination bipartite graphs, and star-convex bipartite graphs. On the positive side, we prove that this problem can be solved in linear time in bipartite chain graphs and polynomial time for trees. Finally, we show three upper bounds for split graphs and give examples of split graphs where the difference between these upper bounds and Trcp(G) can be arbitrarily large. |
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| ISSN: | 0304-3975 |
| DOI: | 10.1016/j.tcs.2025.115582 |