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.

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A graph is ℓ-holed if all its induced cycles of length at least four have length exactly ℓ. We give a complete description of the ℓ-holed graphs for each ℓ ≥ 7.There is a third way to enlarge ℓ-holed graphs to larger ones; choose a vertex and replace it by a set of pairwise adjacent vertices, each with the same neighbours as the original vertex (and one another). If two adjacent vertices have exactly the same neighbours, we call them adjacent twins. We could assume there are no adjacent twins, without loss of generality, but it makes little difference, so usually we will not do so.Some terminology: two disjoint subsets X, Y of a graph are complete if every vertex of X is adjacent to every vertex in Y , and anticomplete if there are no edges between X, Y . Incidentally, we will sometimes use expressions such as "G-adjacent" and "G-neighbour" when we want to make clear which graph we are using, as we will often have different graphs with the same vertex set.Let X, Y be disjoint subsets of V (G); we denote by G[X, Y ] the bipartite subgraph of G with vertex set X ∪ Y and edge set the set of edges of G between X, Y . A half-graph is a bipartite graph with no induced two-edge matching. Thus, a graph with bipartition (X, Y ) is a half-graph if and only if we can order X = {x 1 , . . . , x m } and Y = {y 1 , . . . , y n } such that for all i, i ′ , j, j ′ with 1 ≤ i ≤ i ′ ≤ m and 1 ≤ j ≤ j ′ ≤ n, if x i ′ y j ′ is an edge then x i y j is an edge. Such a half-graph together with the orderings of X, Y is called an ordered half-graph.Half-graphs will be of great importance in this paper. For instance, let G be an ℓ-holed graph where ℓ ≥ 5, and let X, Y be disjoint cliques of G. Since there is no 4-hole with two vertices in X and two inwith first vertex in Y , second and third in X, and fourth in Z, and this latter property is important for keeping all holes the same length. This is also equivalent to asking that there are orderings of X, Y and Z such that G[X, Y ], G[X, Z] are both ordered half-graphs.Let G be a graph with vertex set partitioned into sets W 1 , . . . , W ℓ , with the following properties:

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For every integer ≥ 7, we give a structural description of the class of graphs whose chordless cycles of length at least 4 all have length .