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 NP-complete for various H-free bipartite graphs, e.g., Lu and Tang showed that ED is NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least g for every fixed g. Thus, ED is NP-complete for K 1,4-free bipartite graphs and for C 4-free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for S 1,3,3-free bipartite graphs.

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. 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. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2 +P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3.

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The class L k of k-leaf powers consists of graphs G = (V, E) that have a k-leaf root, that is, a tree T with leaf set V , where xy ∈ E, if and only if the T -distance between x and y is at most k. Structure and linear time recognition algorithms have been found for 2-, 3-, 4-, and, to some extent, 5-leaf powers, and it is known that the union of all k-leaf powers, that is, the graph class L = ∞ k=2 L k , forms a proper subclass of strongly chordal graphs. Despite from that, no essential progress has been made lately.In this paper, we use the new notion of clique arrangements to suggest that leaf powers are a natural special case of strongly chordal graphs. The clique arrangement A(G) of a chordal graph G is a directed graph that represents the intersections between maximal cliques of G by nodes and the mutual inclusion of these vertex subsets by arcs. Recently, strongly chordal graphs have been characterized as the graphs that have a clique arrangement without bad k-cycles for k ≥ 3. We show that the clique arrangement of every graph of L is free of bad 2-cycles. The question whether this characterizes the class L exactly remains open.