Abstract:A graph G has maximal local edge-connectivity k if the maximum number of edge-disjoint paths between every pair of distinct vertices x and y is at most k. We prove Brooks-type theorems for k-connected graphs with maximal local edge-connectivity k, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph G with maximal local connectivity 3, outputs an optimal coloring for G. On the other hand, we prove, for k ≥ 3, that k-COLORABILITY is NP-complete when restricted to minimally k-connected graphs, and 3-COLORABILITY is NPcomplete when restricted to (k − 1)-connected graphs with maximal local connectivity k. Finally, we consider a parameterization of k-COLORABILITY based on the number of vertices of degree at least k + 1, and prove that, even when k is part of the input, the corresponding parameterized problem is FPT. C 2016 Wiley Periodicals, Inc. J. Graph Theory 85: 2017

Abstract. A standard matrix representation of a matroid M represents M relative to a fixed basis B, where contracting elements of B and deleting elements of E(M )−B correspond to removing rows and columns of the matrix, while retaining standard form. If M is 3-connected and N is a 3-connected minor of M , it is often desirable to perform such a removal while maintaining both 3-connectivity and the presence of an N -minor. We prove that, subject to a mild essential restriction, when M has no 4-element fans with a specific labelling relative to B, there are at least two distinct elements, s1 and s2, such that for each i ∈ {1, 2}, si(M/si) is 3-connected and has an N -minor when si ∈ B, and co(M \si) is 3-connected and has an N -minor when si ∈ E(M ) − B. We also show that if M has precisely two such elements and P is the set of elements that, when removed in the appropriate way, retain the N -minor, then (P, E(M ) − P ) is a sequential 3-separation.

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.

It has long been known that Feedback Vertex Set can be solved in time 2 Opw log wq n Op1q on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2 Opwq n Op1q , that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class P of graphs, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G´S has at most d vertices and is in P. Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d:• if P consists only of chordal graphs, then the problem can be solved in time 2 Opwd 2 q n Op1q ,• if P contains a graph with an induced cycle of length ě 4, then the problem is not solvable in time 2 opw log wq n Op1q even for fixed d " , unless the ETH fails.We also study a similar problem, called Bounded P-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if d is part of the input and P contains all chordal graphs, then it cannot be solved in time f pwqn opwq for some function f , unless the ETH fails.Bounded-size components. Using a similar technique, we can obtain analogous results for a simpler problem, which we call Bounded P-Component Vertex Deletion, where we want to remove at most k vertices such that each connected component of the resulting graph has at most d vertices and belongs to P. If we have only the size constraint (i.e., P contains every graph), then this problem is known as Component Order Connectivity [9]. Let P be a class of graphs.Bounded P-Component Vertex Deletion Parameter: d, w Input: A graph G of treewidth at most w, and positive integers d and k. Question: Is there a set S of at most k vertices in G such that each connected component of G´S has at most d vertices and is in P?Drange, Dregi, and van 't Hof [9] studied the parameterized complexity of a weighted variant of the Component Order Connectivity problem; their results imply, in particular, that Component Order Connectivity can be solved in time 2 Opk log dq n, but is W r1s-hard parameterized by only k or d. The corresponding edge-deletion problem, parameterized by treewidth, was studied by Enright and Meeks [10]. For general classes P, we prove results that are analogous to those for Bounded P-Block Vertex Deletion. Theorem 1.3. Let P be a class of graphs that is hereditary, recognizable in polynomial time, and consists of only chordal graphs. Then Bounded P-Component Vertex Deletion can be solved in time 2 Opwd 2 q k 2 n on graphs with n vertices and treewidth w.Theorem 1.4. Let P be a hereditary class of graphs that is polynomial-time recognizable. I...

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