
Thinness of product graphs
The thinness of a graph is a width parameter that generalizes some prope...
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On Efficient Domination for Some Classes of HFree Bipartite Graphs
A vertex set D in a finite undirected graph G is an efficient dominatin...
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Graphs with at most two moplexes
A moplex is a natural graph structure that arises when lifting Dirac's c...
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On the stab number of rectangle intersection graphs
We introduce the notion of stab number and exact stab number of rectangl...
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Extremal solutions to some art gallery and terminalpairability problems
The chosen tool of this thesis is an extremal type approach. The lesson ...
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AxisAligned Square Contact Representations
We introduce a new class 𝒢 of bipartite plane graphs and prove that each...
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Semidefinite Programming in Timetabling and MutualExclusion Scheduling
In scheduling and timetabling applications, the mutualexclusion constra...
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Intersection models and forbidden pattern characterizations for 2thin and proper 2thin graphs
The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NPcomplete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. Proper thinness is defined analogously, generalizing proper interval graphs, and a larger family of NPcomplete problems are known to be polynomially solvable for graphs with bounded proper thinness. It is known that the thinness of a graph is at most its pathwidth plus one. In this work, we prove that the proper thinness of a graph is at most its bandwidth, for graphs with at least one edge. It is also known that boxicity is a lower bound for the thinness. The main results of this work are characterizations of 2thin and 2proper thin graphs as intersection graphs of rectangles in the plane with sides parallel to the Cartesian axes and other specific conditions. We also bound the bend number of graphs with low thinness as vertex intersection graphs of paths on a grid (B_kVPG graphs are the graphs that have a representation in which each path has at most k bends). We show that 2thin graphs are a subclass of B_1VPG graphs and, moreover, of monotone Lgraphs, and that 3thin graphs are a subclass of B_3VPG graphs. We also show that B_0VPG graphs may have arbitrarily large thinness, and that not every 4thin graph is a VPG graph. Finally, we characterize 2thin graphs by a set of forbidden patterns for a vertex order.
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