12 research outputs found
Edge Intersection Graphs of L-Shaped Paths in Grids
In this paper we continue the study of the edge intersection graphs of one
(or zero) bend paths on a rectangular grid. That is, the edge intersection
graphs where each vertex is represented by one of the following shapes:
,, , , and we consider zero bend
paths (i.e., | and ) to be degenerate s. These graphs, called
-EPG graphs, were first introduced by Golumbic et al (2009). We consider
the natural subclasses of -EPG formed by the subsets of the four single
bend shapes (i.e., {}, {,},
{,}, and {,,}) and we
denote the classes by [], [,],
[,], and [,,]
respectively. Note: all other subsets are isomorphic to these up to 90 degree
rotation. We show that testing for membership in each of these classes is
NP-complete and observe the expected strict inclusions and incomparability
(i.e., [] [,],
[,] [,,]
-EPG; also, [,] is incomparable with
[,]). Additionally, we give characterizations and
polytime recognition algorithms for special subclasses of Split
[].Comment: 14 pages, to appear in DAM special issue for LAGOS'1
The Complexity of Helly- EPG Graph Recognition
Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the
intersection graph class of edge paths on a grid. An EPG graph is a graph
that admits a representation where its vertices correspond to paths in a grid
, such that two vertices of are adjacent if and only if their
corresponding paths in have a common edge. If the paths in the
representation have at most bends, we say that it is a -EPG
representation. A collection of sets satisfies the Helly property when
every sub-collection of that is pairwise intersecting has at least one
common element. In this paper, we show that given a graph and an integer
, the problem of determining whether admits a -EPG representation
whose edge-intersections of paths satisfy the Helly property, so-called
Helly--EPG representation, is in NP, for every bounded by a polynomial
function of . Moreover, we show that the problem of recognizing
Helly--EPG graphs is NP-complete, and it remains NP-complete even when
restricted to 2-apex and 3-degenerate graphs
Proper circular arc graphs as intersection graphs of pathson a grid
In this paper we present a characterization, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn).Facultad de Ciencias Exacta
K-SUN PERTENCE A B2-EPG-HELLY
In this article we explore the -EPG class and the Helly property. We present generic results on EPG representations and define terms that support the other results, in addition, we finish the research with an unpublished algorithm that builds a Helly -EPG representation of any k-sun graph.En este artículo exploramos la clase -EPG y la propiedad Helly. Presentamos resultados genéricos sobre las representaciones EPG y definimos términos que respaldan los otros resultados, además, finalizamos la investigación con un algoritmo original que construye una representación -EPG -Helly de cualquier grafo k-sun.Dans cet article, nous explorons la classe de graphes B_2-EPG et la propriété Helly. Nous présentons des résultats génériques sur les représentations EPG et définissons les termes qui supportent les autres résultats, en plus, nous présentons un algorithme sans précédent qui construit une représentation B_2-EPG-Helly de tout graphe k-sun.Neste artigo exploramos a classe de grafos -EPG e a propriedade Helly. Apresentamos resultados genéricos sobre representações EPG e definimos termos que suportam os demais resultados, além disso, apresentamos um algoritmo inédito que constrói uma representação -EPG-Helly de qualquer grafo k-sun
Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II
Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT
graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the
EPT graph (i.e. the edge intersection graph) of P. These graphs have been
extensively studied in the literature. Given two (edge) intersecting paths in a
graph, their split vertices is the set of vertices having degree at least 3 in
their union. A pair of (edge) intersecting paths is termed non-splitting if
they do not have split vertices (namely if their union is a path). We define
the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed
the ENPT graph, as the graph having a vertex for each path in P, and an edge
between every pair of vertices representing two paths that are both
edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a
tree T and a set of paths P of T such that G=ENPT(P), and we say that is
a representation of G.
Our goal is to characterize the representation of chordless ENPT cycles
(holes). To achieve this goal, we first assume that the EPT graph induced by
the vertices of an ENPT hole is given. In [2] we introduce three assumptions
(P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we
define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize
the representations of ENPT holes that satisfy (P1), (P2), (P3).
In this work, we continue our work by relaxing these three assumptions one by
one. We characterize the representations of ENPT holes satisfying (P3) by
providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also
show that there does not exist a polynomial-time algorithm to solve
HamiltonianPairRec, unless P=NP