Reducing Graph Transversals via Edge Contractions

For a graph parameter ?, the Contraction(?) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which ? has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where ? is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ? according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ?, which in particular imply that Contraction(?) is co-NP-hard even for fixed k = d = 1 when ? is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when ? is the size of a minimum vertex cover, the problem is in XP parameterized by d

Three problems on well-partitioned chordal graphs

In this work, we solve three problems on well-partitioned chordal graphs. First, we show that every connected (resp., 2-connected) well-partitioned chordal graph has a vertex that intersects all longest paths (resp., longest cycles). It is an open problem [Balister et al., Comb. Probab. Comput. 2004] whether the same holds for chordal graphs. Similarly, we show that every connected well-partitioned chordal graph admits a (polynomial-time constructible) tree 3-spanner, while the complexity status of the Tree 3-Spanner problem remains open on chordal graphs [Brandstädt et al., Theor. Comput. Sci. 2004]. Finally, we show that the problem of finding a minimum-size geodetic set is polynomial-time solvable on well-partitioned chordal graphs. This is the first example of a problem that is NP -hard on chordal graphs and polynomial-time solvable on well-partitioned chordal graphs. Altogether, these results reinforce the significance of this recently defined graph class as a tool to tackle problems that are hard or unsolved on chordal graphs.acceptedVersio

On Blockers and Transversals of Maximum Independent Sets in Co-Comparability Graphs

In this paper, we consider the following two problems: (i) Deletion Blocker($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that $\alpha(G-S) \leq \alpha(G)-d$, that is the independence number of $G$ decreases by at least $d$ after having removed the vertices from $S$; (ii) Transversal($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that for every maximum independent set $I$ we have $|I\cap S| \geq d$. We show that both problems are polynomial-time solvable in the class of co-comparability graphs by reducing them to the well-known Vertex Cut problem. Our results generalize a result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al., Assistance and interdiction problems on interval graphs, Discrete Applied Mathematics 340, pp. 153-170, 2023]

Sublinear Longest Path Transversals and Gallai Families

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit constant-size longest path transversals. The same technique allows us to show that $2$-connected graphs admit sublinear longest cycle transversals. We also make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. Finally, we show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$ and $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$

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)|

The intersection of longest paths in a graph.

In this thesis we examine the famous conjecture that every three longest paths in a graph intersect, and add to the classes of graphs for which it is known that this conjecture holds. This conjecture arose from a question asked by Gallai in 1966, the question of whether all of the longest paths in a graph intersect (Gallai's question). In 1969, Walther found a graph in which the longest paths do not all intersect, answering Gallai's question. Since then, many other graphs in which the longest paths do not all intersect have been found. However there are also many classes of graphs for which the longest paths all intersect, such as series-parallel graphs and dually chordal graphs. Finding such classes of graphs is an active area of research and in this thesis we add to these classes of graphs. We begin by investigating Gallai's question for a speci c class of graphs. A theta graph is a graph consisting of three paths with a pair of common endpoints and no other common vertices. A generalised theta graph is a graph with at least one block that consists of at least three paths with a pair of common endpoints and no other common vertices. We show that for a subclass of generalised theta graphs, all of the longest paths intersect. Next, we consider the conjecture that every three longest paths of a graph intersect. We prove that, for every graph with n vertices and at most n + 5 edges, every three longest paths intersect. Finally, we use computational methods to investigate whether all longest paths intersect, or every three longest paths intersect, for several classes of graphs. Two graphs are homeomorphic if each can be obtained from the same graph H by a series of subdivisions. We show that, for every simple connected graph G that is homeomorphic to a simple connected graph with at most 7 vertices, all of the longest paths of G intersect. Additionally, we show that, for every simple connected graph G homeomorphic to a simple connected graph with n vertices, n + 6 edges, and minimum vertex degree 3, all of the longest paths of G intersect. We then show that for every graph with n vertices and at most n + 5 edges, every three longest paths intersect, independently verifying this result. We also present results for several additional classes of graphs with conditions on the blocks, maximum degree of the vertices, and other properties of the graph, showing that every three longest paths intersect or every six longest paths intersect for these graphs