We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: “globally noncrossing” graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the ...

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprisin...

We examine the simultaneous embedding with fixed edges problem for two planar graphs G1 and G2 with the focus on their intersection S = G1∩G2. In particular, we will present the complete set of intersection graphs S that guarantee a simultaneous embedding with fixed edges for (G1, G2). More formally, we define the subset ISEFE of all planar graphs as follows: A graph S lies in ISEFE if every pa...

We prove that the unique decomposition of connected graphs defined by Tutte is definable by formulas of Monadic Second-Order Logic. This decomposition has two levels: every connected graph is a tree of "2-connected components" called blocks ; every 2-connected graph is a tree of so-called 3-blocks. Our proof uses 2dags which are certain acyclic orientations of the considered graphs. We obtain a...

The rank-width is a graph parameter related in terms of fixed functions to cliquewidth but more tractable. Clique-width has nice algorithmic properties, but no good “minor” relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartit...

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The popularity of Agile Development has been increasing over the last several years. Many Agile techniques include unit testing as a basic activity. Unit testing has two characteristics: it is performed by programmers, and it is normally based on source code structures. However, the problem arises that programmers are unlikely to design structure-based tests. Test design process draws graphs th...

A universal representation theorem is derived that shows any graph is the intersection graph of one chordal graph, a number of co-bipartite graphs, and one unit interval graph. Central to the the result is the notion of the clique cover width which is a generalization of the bandwidth parameter. Specifically, we show that any planar graph is the intersection graph of one chordal graph, four co-...