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In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0-CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG.

A semitotal dominating set of a graph G with no isolated vertex is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The minimum size γ t2 (G) of a semitotal dominating set of G is squeezed between the domination number γ(G) and the total domination number γ t (G).SEMITOTAL DOMINATING SET is the problem of finding, given a graph G, a semitotal dominating set of G of size γ t2 (G). In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph classes. In particular, we show that it is solvable in polynomial time for the class of graphs with bounded mim-width by a reduction to TOTAL DOMINATING SET and we provide several approximation lower bounds for subclasses of subcubic graphs. Moreover, we obtain complexity dichotomies in monogenic classes for the decision versions of SEMITOTAL DOMINATING SET and TOTAL DOMINATING SET.Finally, we show that it is NP-complete to recognise the graphs such that γ t2 (G) = γ t (G) and those such that γ(G) = γ t2 (G), even if restricted to be planar and with maximum degree at most 4, and we provide forbidden induced subgraph characterisations for the graphs heriditarily satisfying either of these two equalities.

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