Related papers: Extending Partial Representations of Interval Grap…
The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire…
Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some…
The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle…
Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time.…
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition…
The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is…
The interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each intersecting pair of intervals. A probe interval graph is a variant that is motivated by an application to genomics, where…
In a recent paper, we introduced the simultaneous representation problem (defined for any graph class C) and studied the problem for chordal, comparability and permutation graphs. For interval graphs, the problem is defined as follows. Two…
We introduce the class of interval $H$-graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed graph $H$ with vertices $a_1,a_2,\dots,a_k$, we say that an input graph $G$ with given partition…
Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T' and some pre-drawn subtrees of…
Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of…
The partial representation extension problem, introduced by Klav\'{i}k et al. (2011), generalizes the recognition problem. In this short note we show that this problem is NP-complete for unit circular-arc graphs.
Let $\mbox{interval} + k v$, $\mbox{interval} + k e$, and $\mbox{interval} - k e$ denote the classes of graphs that can be obtained from some interval graph by adding $k$ vertices, adding $k$ edges, and deleting $k$ edges, respectively.…
In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called $k$-length interval graphs were considered in which the number of different lengths…
A rectangular dual of a graph $G$ is a contact representation of $G$ by axis-aligned rectangles such that (i)~no four rectangles share a point and (ii)~the union of all rectangles is a rectangle. The partial representation extension problem…
The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$,…
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where…
Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a…
Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the…
Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure…