193 research outputs found
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)|
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 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
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
Robust Hamiltonicity in families of Dirac graphs
A graph is called Dirac if its minimum degree is at least half of the number
of vertices in it. Joos and Kim showed that every collection
of Dirac graphs on the same vertex set of
size contains a Hamilton cycle transversal, i.e., a Hamilton cycle on
with a bijection such that
for every .
In this paper, we determine up to a multiplicative constant, the threshold
for the existence of a Hamilton cycle transversal in a collection of random
subgraphs of Dirac graphs in various settings. Our proofs rely on constructing
a spread measure on the set of Hamilton cycle transversals of a family of Dirac
graphs.
As a corollary, we obtain that every collection of Dirac graphs on
vertices contains at least different Hamilton cycle transversals
for some absolute constant . This is optimal up to the constant
. Finally, we show that if is sufficiently large, then every such
collection spans pairwise edge-disjoint Hamilton cycle transversals, and
this is best possible. These statements generalize classical counting results
of Hamilton cycles in a single Dirac graph
On two conjectures about the intersection of longest paths and cycles
A conjecture attributed to Smith states that every pair of longest cycles in
a -connected graph intersect each other in at least vertices. In this
paper, we show that every pair of longest cycles in a~-connected graph on
vertices intersect each other in at least~ vertices,
which confirms Smith's conjecture when . An analog conjecture
for paths instead of cycles was stated by Hippchen. By a simple reduction, we
relate both conjectures, showing that Hippchen's conjecture is valid when
either or
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