A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration.We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied.We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H1, H2)-free graphs. We identify several general classes of (H1, H2)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H1, H2)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H1 and H2, the class of (H1, H2)-free graphs has unbounded mim-width.Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H1, H2)-free graphs. In particular, we classify the mim-width of (H1, H2)-free graphs for all pairs (H1, H2) with |V (H1)| + |V (H2)| ≤ 8. When H1 and H2 are connected graphs, we classify all pairs (H1, H2) except for one remaining infinite family and a few isolated cases.

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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