203,577 research outputs found
Completion of the mixed unit interval graphs hierarchy
We describe the missing class of the hierarchy of mixed unit interval graphs,
generated by the intersection graphs of closed, open and one type of half-open
intervals of the real line. This class lies strictly between unit interval
graphs and mixed unit interval graphs. We give a complete characterization of
this new class, as well as quadratic-time algorithms that recognize graphs from
this class and produce a corresponding interval representation if one exists.
We also mention that the work in arXiv:1405.4247 directly extends to provide a
quadratic-time algorithm to recognize the class of mixed unit interval graphs.Comment: 17 pages, 36 figures (three not numbered). v1 Accepted in the TAMC
2015 conference. The recognition algorithm is faster in v2. One graph was not
listed in Theorem 7 of v1 of this paper v3 provides a proposition to
recognize the mixed unit interval graphs in quadratic time. v4 is a lot
cleare
A Characterization of Mixed Unit Interval Graphs
We give a complete characterization of mixed unit interval graphs, the
intersection graphs of closed, open, and half-open unit intervals of the real
line. This is a proper superclass of the well known unit interval graphs. Our
result solves a problem posed by Dourado, Le, Protti, Rautenbach and
Szwarcfiter (Mixed unit interval graphs, Discrete Math. 312, 3357-3363 (2012)).Comment: 17 pages, referees' comments adde
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
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