Longest Path and Cycle Transversal and Gallai Families

A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and is denoted lct(G). A Gallai family is a family of graphs whose connected members have longest path transversal number 1. In this paper we find several Gallai families and give upper bounds on lpt(G) and lct(G) for general graphs and chordal graphs in terms of |V(G)|

Graph Transversals for Hereditary Graph Classes: a Complexity Perspective

Within the broad field of Discrete Mathematics and Theoretical Computer Science, the theory of graphs has been of fundamental importance in solving a large number of optimization problems and in modelling real-world situations. In this thesis, we study a topic that covers many aspects of Graph Theory: transversal sets. A transversal set in a graph G is a vertex set that intersects every subgraph of G that belongs to a certain class of graphs. The focus is on vertex cover, feedback vertex set and odd cycle transversal. The decision problems Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal ask, for a given graph G and an integer k, whether there is a corresponding transversal of G of size at most k. These problems are NP-complete in general and our focus is to determine the complexity of the problems when various restrictions are placed on the input, both for the purpose of finding tractable cases and to increase our understanding of the point at which a problem becomes NP-complete. We consider graph classes that are closed under vertex deletion and in particular H-free graphs, i.e. graphs that do not contain a graph H as an induced subgraph. The first chapter is an introduction to the thesis. There we illustrate the motivation of our work and introduce most of the terminology we have used for our research. In the second chapter, we develop a number of structural results for some classes of H-free graphs. The third chapter looks at the Subset Transversal problems: there we prove that Feedback Vertex Set and Odd Cycle Transversal and their subset variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs, while for Subset Vertex Cover we show that it can be solved in polynomial time for (sP_1+P_4)-free graphs. The fourth chapter is entirely dedicated to the Connected Vertex Cover problem. The connectivity constraint requires additional proof techniques. We prove this problem can be solved in polynomial time for (sP_1+P_5)-free graphs, even when weights are given to the vertices of the graph. We continue the research on connected transversals in the fifth chapter: we show that Connected Feedback Vertex Set, Connected Odd Cycle Transversal and their extension variants can be solved in polynomial time for both P_4-free and (sP_1+P_3)-free graphs. In the sixth chapter we study the price of independence: can the size of a smallest independent transversal be bounded in terms of the minimum size of a transversal? We establish complete and almost-complete dichotomies which determine for which graph classes such a bound exists and for which cases such a bound is the identity

Application of hypergraphs in decomposition of discrete systems

seria: Lecture Notes in Control and Computer Science ; vol. 23

Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal G}$. We let ${\cal G}$ be the class of $k$-degenerate graphs. This problem is known to be polynomial-time solvable if $k=0$ (bipartite graphs) and NP-complete if $k=1$ (near-bipartite graphs) even for graphs of maximum degree $4$. Yang and Yuan [DM, 2006] showed that the $k=1$ case is polynomial-time solvable for graphs of maximum degree $3$. This also follows from a result of Catlin and Lai [DM, 1995]. We consider graphs of maximum degree $k+2$ on $n$ vertices. We show how to find $A$ and $B$ in $O(n)$ time for $k=1$, and in $O(n^2)$ time for $k\geq 2$. Together, these results provide an algorithmic version of a result of Catlin [JCTB, 1979] and also provide an algorithmic version of a generalization of Brook's Theorem, which was proven in a more general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007]. Moreover, the two results enable us to complete the complexity classification of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex colouring reconfiguration graph between two given $\ell$-colourings of a graph of maximum degree $k$

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

Contraction blockers for graphs with forbidden induced paths.

We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pℓ-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs

Disjoint list-colorings for planar graphs

One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find $3$ disjoint $L$-colorings (a packing), or $2$ disjoint $L$-colorings, or a collection of $L$-colorings that to every vertex assigns every color on average in one third of the cases (a fractional packing)? We prove that the packing is impossible, but two disjoint $L$-colorings are guaranteed if the girth is at least $8$, and a fractional packing exists when the girth is at least $6.$ For a graph $G$, the least $k$ such that there are always $k$ disjoint proper list-colorings whenever we have lists all of size $k$ associated to the vertices is called the list packing number of $G$. We lower the two-times-degeneracy upper bound for the list packing number of planar graphs of girth $3,4$ or $5$. As immediate corollaries, we improve bounds for $\epsilon$-flexibility of classes of planar graphs with a given girth. For instance, where previously Dvo\v{r}\'{a}k et al. proved that planar graphs of girth $6$ are (weighted) $\epsilon$-flexibly $3$-choosable for an extremely small value of $\epsilon$, we obtain the optimal value $\epsilon=\frac{1}{3}$. Finally, we completely determine and show interesting behavior on the packing numbers for $H$-minor-free graphs for some small graphs $H.$Comment: 36 pages, 8 figure