592 research outputs found
Linear Time Recognition of P4-Indifferent Graphs
A simple graph is P4-indifferent if it admits a total order b > c > d. P4-indifferent graphs generalize indifferent graphs and are perfectly orderable. Recently, Hoang,Maray and Noy gave a characterization of P4-indifferent graphs interms of forbidden induced subgraphs. We clarify their proof and describe a linear time algorithm to recognize P4-indifferent graphs. Whenthe input is a P4-indifferent graph, then the algorithm computes an order < as above.Key words: P4-indifference, linear time, recognition, modular decomposition.
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
Intersection representation of digraphs in trees with few leaves
The leafage of a digraph is the minimum number of leaves in a host tree in
which it has a subtree intersection representation. We discuss bounds on the
leafage in terms of other parameters (including Ferrers dimension), obtaining a
string of sharp inequalities.Comment: 12 pages, 3 included figure
Maximizing the Strong Triadic Closure in Split Graphs and Proper Interval Graphs
In social networks the Strong Triadic Closure is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of strong edges that satisfy the strong triadic closure was recently shown to be NP-complete for general graphs. Here we initiate the study of graph classes for which the problem is solvable. We show that the problem admits a polynomial-time algorithm for two unrelated classes of graphs: proper interval graphs and trivially-perfect graphs. To complement our result, we show that the problem remains NP-complete on split graphs, and consequently also on chordal graphs. Thus we contribute to define the first border between graph classes on which the problem is polynomially solvable and on which it remains NP-complete
Mathematical Properties on the Hyperbolicity of Interval Graphs
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.This paper was supported in part by a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), MĂ©xico and by two grants from the Ministerio de EconomĂa y Competitividad, Agencia Estatal de InvestigaciĂłn (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain. We would like to thank the referees for their careful reading of the manuscript and several useful comments that have helped us to improve the presentation of the paper
Preferences on Intervals: a general framework
I present a general framework for the comparison of alternatives to which (possibly) an interval of values is associated. Some representation theorems for the existence of the intervals are discussed as well the possibility ot explicitly take into account situations of hesitation.
Some appropriate logical formalisms are discussed for such a purpose
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