Constrained Representations of Map Graphs and Half-Squares

Hoang-Oanh Le, Van Bang Le, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
The square of a graph H, denoted H 2 , is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B = (X, Y, EB) are the subgraphs of B 2 induced by the color classes X and Y , B 2 [X] and While Chen, Grigni, Papadimitriou proved that any map graph G = (V, EG) has a witness with at most 3|V | − 6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is
more » ... P-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown. We also consider half-squares of tree-convex bipartite graphs and prove the following complexity dichotomy: Given a graph G = (V, EG) and an integer k, deciding if G = B 2 [V ] for some tree-convex bipartite graph B = (V, W, EB) with |W | ≤ k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs.
doi:10.4230/lipics.mfcs.2019.13 dblp:conf/mfcs/LeL19 fatcat:lyte7e2k7rhc5pfgfqfdmkz7py