Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

Graph Partitioning With Input Restrictions

In this thesis we study the computational complexity of a number of graph partitioning problems under a variety of input restrictions. Predominantly, we research problems related to Colouring in the case where the input is limited to hereditary graph classes, graphs of bounded diameter or some combination of the two. In Chapter 2 we demonstrate the dramatic eect that restricting our input to hereditary graph classes can have on the complexity of a decision problem. To do this, we show extreme jumps in the complexity of three problems related to graph colouring between the class of all graphs and every other hereditary graph class. We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be H-free for some graph H if it contains no induced subgraph isomorphic to H. Similarly, G is said to be H-free for some set of graphs H, if it does not contain any graph in H as an induced subgraph. Here, the set H consists usually of a single cycle or tree but may also contain a number of cycles, for example we give results for graphs of bounded diameter and girth. Chapter 4 is dedicated to three variants of the Colouring problem, Acyclic Colouring, Star Colouring, and Injective Colouring. We give complete or almost complete dichotomies for each of these decision problems restricted to H-free graphs. In Chapter 5 we study these problems, along with three further variants of 3-Colouring, Independent Odd Cycle Transversal, Independent Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of bounded diameter. Finally, Chapter 6 deals with a dierent variety of problems. We study the problems Disjoint Paths and Disjoint Connected Subgraphs for H-free graphs

Minors and dimension

It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference

Three ways to cover a graph

We consider the problem of covering an input graph $H$ with graphs from a fixed covering class $G$. The classical covering number of $H$ with respect to $G$ is the minimum number of graphs from $G$ needed to cover the edges of $H$ without covering non-edges of $H$. We introduce a unifying notion of three covering parameters with respect to $G$, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures "how far'' $H$ is from $G$ in a different way. Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention. We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity. As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs. For several pairs of an input class and a covering class we determine exactly the maximum ordinary, local, and folded covering number of an input graph with respect to that covering class.Comment: 20 pages, 4 figure