5,532 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
Clique coloring -EPG graphs
We consider the problem of clique coloring, that is, coloring the vertices of
a given graph such that no (maximal) clique of size at least two is
monocolored. It is known that interval graphs are -clique colorable. In this
paper we prove that -EPG graphs (edge intersection graphs of paths on a
grid, where each path has at most one bend) are -clique colorable. Moreover,
given a -EPG representation of a graph, we provide a linear time algorithm
that constructs a -clique coloring of it.Comment: 9 Page
Polynomial sequences of binomial-type arising in graph theory
In this paper, we show that the solution to a large class of "tiling"
problems is given by a polynomial sequence of binomial type. More specifically,
we show that the number of ways to place a fixed set of polyominos on an
toroidal chessboard such that no two polyominos overlap is
eventually a polynomial in , and that certain sets of these polynomials
satisfy binomial-type recurrences. We exhibit generalizations of this theorem
to higher dimensions and other lattices. Finally, we apply the techniques
developed in this paper to resolve an open question about the structure of
coefficients of chromatic polynomials of certain grid graphs (namely that they
also satisfy a binomial-type recurrence).Comment: 15 page
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
Clique coloring B1-EPG graphs
We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable.
Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.Facultad de Ciencias Exacta
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