12 research outputs found
Longest Paths in Circular Arc Graphs
As observed by Rautenbach and Sereni (arXiv:1302.5503) there is a gap in the
proof of the theorem of Balister et al. (Longest paths in circular arc graphs,
Combin. Probab. Comput., 13, No. 3, 311-317 (2004)), which states that the
intersection of all longest paths in a connected circular arc graph is
nonempty. In this paper we close this gap.Comment: 7 page
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
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
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 -connected
graphs admit sublinear longest cycle transversals.
We also make progress toward a characterization of the graphs such that
every connected -free graph has a longest path transversal of size . In
particular, we show that the graphs on at most vertices satisfying this
property are exactly the linear forests.
Finally, we show that if the order of a connected graph is large relative
to its connectivity and , then each
vertex of maximum degree forms a longest path transversal of size
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)|
Hypohamiltonian and almost hypohamiltonian graphs
This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results