We study a variant of intersection representations with unit balls: unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for two adjacent vertices intersect if and only if the corresponding edge is near. We consider ...

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph $\mathbb{T}$-gain $\Phi$. Similarly to what happens in context signed graphs, real number $\min Spec(A(\mathcal (\Phi))$, that is, smallest eigenvalue adjacency matrix L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring (\Phi))=-2$ are identified. When such ful...

A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic Breadth-First Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the so-called Maximal Neighborhood Search (MNS) and show...

A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations. An graph is an interval graph if its vertices can be assigned intervals on the real line so that vertices are adjacent if and only if the corresponding intervals intersect; such an assignment is an interval representation. When the intervals have the same le...

A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labelled unit disk and disk graphs on n vertices. We show that the number of unit disk graphs on n vertices is n · α(n) and the number of disk graphs on n vertices is n · β(n), where α(n) and β(n) are Θ(1...

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time....

We show that the class of unit grid intersection graphs properly includes both of the classes of interval bigraphs and of P6-free chordal bipartite graphs. We also demonstrate that the classes of unit grid intersection graphs and of chordal bipartite graphs are incomparable. © 2007 Elsevier B.V. All rights reserved.

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