364 research outputs found
Lict edge semientire graph of a planar graph.
In this paper, we introduce the concept of the Lict edge semientire graph of a planar graph. We present characterizations of graphs whose lict edge semientire graphs are planar, outerplanar and Maximal outerplanar, crossing number one. Further, we establish a characterization of graphs whose lict edge semi entire graphs are Eularian and Hamiltonian
Structure and properties of maximal outerplanar graphs.
Outerplanar graphs are planar graphs that have a plane embedding in which each vertex lies on the boundary of the exterior region. An outerplanar graph is maximal outerplanar if the graph obtained by adding an edge is not outerplanar. Maximal outerplanar graphs are also known as triangulations of polygons. The spine of a maximal outerplanar graph G is the dual graph of G without the vertex that corresponds to the exterior region. In this thesis we study metric properties involving geodesic intervals, geodetic sets, Steiner sets, different concepts of boundary, and also relationships between the independence numbers and domination numbers of maximal outerplanar graphs and their spines. In Chapter 2 we find an extension of a result by Beyer, et al. [3] that deals with Hamiltonian degree sequences in maximal outerplanar graphs. In Chapters 3 and 4 we give sharp bounds relating the independence number and domination number, respectively, of a maximal outerplanar graph to those of its spine. In Chapter 5 we discuss the boundary, contour, eccentricity, periphery, and extreme set of a graph. We give a characterization of the boundary of maximal outerplanar graphs that involves the degrees of vertices. We find properties that characterize the contour of a maximal outerplanar graph. The other main result of this chapter gives characterizations of graphs induced by the contour and by the periphery of a maximal outerplanar graph. In Chapter 6 we show that the generalized intervals in a maximal outerplanar graph are convex. We use this result to characterize geodetic sets in maximal outerplanar graphs. We show that every Steiner set in a maximal outerplanar graph is a geodetic set and also show some differences between these types of sets. We present sharp bounds for geodetic numbers and Steiner numbers of maximal outerplanar graphs
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
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
Definability Equals Recognizability for -Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph
property that can be defined by a sentence in counting monadic second order
logic (CMSOL) can be checked in linear time for graphs of bounded treewidth,
which is known as Courcelle's Theorem. These algorithms are constructed as
finite state tree automata, and hence every CMSOL-definable graph property is
recognizable. Courcelle also conjectured that the converse holds, i.e. every
recognizable graph property is definable in CMSOL for graphs of bounded
treewidth. We prove this conjecture for -outerplanar graphs, which are known
to have treewidth at most .Comment: 40 pages, 8 figure
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