29 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)|
Domination and Cut Problems on Chordal Graphs with Bounded Leafage
The leafage of a chordal graph G is the minimum integer l such that G can berealized as an intersection graph of subtrees of a tree with l leaves. Weconsider structural parameterization by the leafage of classical domination andcut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018,Algorithmica 2020] proved, among other things, that Dominating Set on chordalgraphs admits an algorithm running in time . We present aconceptually much simpler algorithm that runs in time . Weextend our approach to obtain similar results for Connected Dominating Set andSteiner Tree. We then consider the two classical cut problems MultiCut withUndeletable Terminals and Multiway Cut with Undeletable Terminals. We provethat the former is W[1]-hard when parameterized by the leafage and complementthis result by presenting a simple -time algorithm. To our surprise,we find that Multiway Cut with Undeletable Terminals on chordal graphs can besolved, in contrast, in -time.<br
Domination and Cut Problems on Chordal Graphs with Bounded Leafage
The leafage of a chordal graph is the minimum integer such that can be realized as an intersection graph of subtrees of a tree with leaves.
We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs.
Fomin, Golovach, and Raymond~[ESA~, Algorithmica~] proved, among other things, that \textsc{Dominating Set} on chordal graphs admits an algorithm running in time .
We present a conceptually much simpler algorithm that runs in time .
We extend our approach to obtain similar results for \textsc{Connected Dominating Set} and \textsc{Steiner Tree}.
We then consider the two classical cut problems \textsc{MultiCut with Undeletable Terminals} and \textsc{Multiway Cut with Undeletable Terminals}.
We prove that the former is \textsf{W}[1]-hard when parameterized by the leafage and complement this result by presenting a simple -time algorithm.
To our surprise, we find that \textsc{Multiway Cut with Undeletable Terminals} on chordal graphs can be solved, in contrast, in -time
Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs
In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph
, the class of -graphs, defined as the intersection graphs of connected
subgraphs of some subdivision of . Recently, quite a lot of research has
been devoted to understanding the tractability border for various computational
problems, such as recognition or isomorphism testing, in classes of -graphs
for different graphs . In this work we undertake this research topic,
focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and
Zeman showed, for every fixed tree , a polynomial-time algorithm recognizing
-graphs. Tucker showed a polynomial time algorithm recognizing -graphs
(circular-arc graphs). On the other hand, Chaplick at al. showed that
recognition of -graphs is -hard if contains two different cycles
sharing an edge.
The main two results of this work narrow the gap between the -hard and
cases of -graphs recognition. First, we show that recognition of
-graphs is -hard when contains two different cycles. On the other
hand, we show a polynomial-time algorithm recognizing -graphs, where is
a graph containing a cycle and an edge attached to it (-graphs are called
lollipop graphs). Our work leaves open the recognition problems of -graphs
for every unicyclic graph different from a cycle and a lollipop. Other
results of this work, which shed some light on the cases that remain open, are
as follows. Firstly, the recognition of -graphs, where is a fixed
unicyclic graph, admits a polynomial time algorithm if we restrict the input to
graphs containing particular holes (hence recognition of -graphs is probably
most difficult for chordal graphs). Secondly, the recognition of medusa graphs,
which are defined as the union of -graphs, where runs over all unicyclic
graphs, is -complete
Obstructions to Faster Diameter Computation: Asteroidal Sets
Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every -edge graph in can be computed in deterministic time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive -approximation of all vertex eccentricities in deterministic time. This is in sharp contrast with general -edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in time for any . As important special cases of our main result, we derive an -time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an -time algorithm for this problem on graphs of asteroidal number at most . We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions
On weighted clique graphs
Let K(G) be the clique graph of a graph G. A m-weighting of K(G) consists on giving to each m-size subset of its vertices a weight equal to the size of the intersection of the m corresponding cliques of G. The 2-weighted clique graph was previously considered by McKee. In this work we obtain a characterization of weighted clique graphs similar to Roberts and Spencer’s characterization for clique graphs.
Some graph classes can be naturally defined in terms of their weighted clique graphs, for example clique-Helly graphs and their generalizations, and diamond-free graphs. The main contribution of this work is to characterize several graph classes by means of their weighted clique graph: hereditary clique-Helly graphs, split graphs, chordal graphs, UV graphs, interval graphs, proper interval graphs, trees, and block graphs.Sociedad Argentina de Informática e Investigación Operativ
Graphs with at most two moplexes
A moplex is a natural graph structure that arises when lifting Dirac's
classical theorem from chordal graphs to general graphs. However, while every
non-complete graph has at least two moplexes, little is known about structural
properties of graphs with a bounded number of moplexes. The study of these
graphs is motivated by the parallel between moplexes in general graphs and
simplicial modules in chordal graphs: Unlike in the moplex setting, properties
of chordal graphs with a bounded number of simplicial modules are well
understood. For instance, chordal graphs having at most two simplicial modules
are interval. In this work we initiate an investigation of -moplex graphs,
which are defined as graphs containing at most moplexes. Of particular
interest is the smallest nontrivial case , which forms a counterpart to
the class of interval graphs. As our main structural result, we show that the
class of connected -moplex graphs is sandwiched between the classes of
proper interval graphs and cocomparability graphs; moreover, both inclusions
are tight for hereditary classes. From a complexity theoretic viewpoint, this
leads to the natural question of whether the presence of at most two moplexes
guarantees a sufficient amount of structure to efficiently solve problems that
are known to be intractable on cocomparability graphs, but not on proper
interval graphs. We develop new reductions that answer this question negatively
for two prominent problems fitting this profile, namely Graph Isomorphism and
Max-Cut. On the other hand, we prove that every connected -moplex graph
contains a Hamiltonian path, generalising the same property of connected proper
interval graphs. Furthermore, for graphs with a higher number of moplexes, we
lift the previously known result that graphs without asteroidal triples have at
most two moplexes to the more general setting of larger asteroidal sets
The Neighborhood Polynomial of Chordal Graphs
The neighborhood polynomial of a graph is the generating function of
subsets of vertices in that have a common neighbor. In this paper we study
the neighborhood polynomial and the complexity of its computation for chordal
graphs. We will show that it is \NP-hard to compute the neighborhood polynomial
on general chordal graphs. Furthermore we will introduce a parameter for
chordal graphs called anchor width and an algorithm to compute the neighborhood
polynomial which runs in polynomial time if the anchor width is polynomially
bounded. Finally we will show that we can bound the anchor width for chordal
comparability graphs and chordal graphs with bounded leafage. The leafage of a
chordal graphs is the minimum number of leaves in the host tree of a subtree
representation. In particular, interval graphs have leafage at most 2. This
shows that the anchor width of interval graphs is at most quadratic