The P4 is the induced path of four vertices. The gem consists of a P4 with an additional universal vertex being completely adjacent to the P4, and the co-gem is its complement graph. Gem- and co-gem-free graphs generalize the popular class of cographs (i. e. P4-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of clique-width of graphs which extends the famous concept of treewidth. It is well-known that the cographs are exactly those graphs of clique-width at most 2. In this paper, we show that the clique-width of gem- and co-gem-free graphs is at most 16.

Let G be a finite undirected graph with edge set E. An edge set E ′ ⊆ E is an induced matching in G if the pairwise distance of the edges of E ′ in G is at least two; E ′ is dominating in G if every edge e ∈ E \ E ′ intersects some edge in E ′ . The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E ′ which is also dominating in G; this problem is also known as the Efficient Edge Domination Problem.The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for P k -free graphs for any k ≥ 5; P k denotes a chordless path with k vertices and k − 1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P 7 -free graphs in a robust way.

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