On Efficient Domination for Some Classes of H-Free Bipartite Graphs
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A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for 2P_3-free chordal graphs while it is solvable in polynomial time for P_6-free chordal graphs (and even for P_6-free graphs). On the other hand, ED is solvable in linear time for 2P_2-free and even for P_5-free graphs, and thus for P_5-free bipartite graphs. Lu and Tang showed that ED is NP-complete for chordal bipartite graphs and for planar bipartite graphs;actually, ED is -complete even for planar bipartite graphs with vertex degree at most 3. Thus, ED is NP-complete for K_1,4-free bipartite graphs. For K_1,3-free bipartite graphs, however, ED is solvable in polynomial time: For classes of bounded clique-width, ED is solvable in polynomial time. Dabrowski and Paulusma published a dichotomy for clique-width of H-free bipartite graphs. For instance, clique-width of S_1,2,3-free bipartite graphs is bounded (which includes K_1,3-free bipartite graphs). We show that ED is NP-complete for K_3,3-free bipartite graphs. Moreover, we show that (weighted) ED can be solved in polynomial time for H-free bipartite graphs when H is P_7 or S_1,2,4 or ℓ P_4 for fixed ℓ, and similarly for P_9-free bipartite graphs with vertex degree at most 3.
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