Tolerance graphs have been extensively studied since their introduction in 1982 [9], due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs [10]. The conjecture has been proved...

We generalise the family of (σ, ρ)-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-r dominating set and distance-r independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and ...

We study powers of certain geometric intersection graphs: interval graphs, m-trapezoid graphs and circular-arc graphs. We define the pseudo product, (G, G)→ G∗G, of two graphs G and G on the same set of vertices, and show that G∗G is contained in one of the three classes of graphs mentioned here above, if both G and G are also in that class and fulfill certain conditions. This gives a new proof...

We make an attempt to understand the dominating set counting problem in graph classes from the viewpoint of polynomial-time computability. We give polynomial-time algorithms to count the number of dominating sets (and minimum dominating sets) in interval graphs and trapezoid graphs. They are based on dynamic programming. With the help of dynamic update on a binary tree, we further reduce the ti...

A circle n-gon is the region between n non-crossing chords of a circle; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The fa...

A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists an edgeweighted tree T and two non-negative real numbers dmin and dmax such that each leaf u of T is a node of V and there is an edge (u, v) ∈ E if and only if dmin ≤ dT (u, v) ≤ dmax where dT (u, v) is the sum of weights of the edges on the unique path from u to v in T . The main issue on these graphs consists in charac...

Abstract. In this paper we show the nonequivalence of combinations of several natural geometric restrictions on trapezoid representations of trapezoid orders. Each of the properties unit parallelogram, unit trapezoid and proper parallelogram, unit trapezoid and parallelogram, unit trapezoid, proper parallelogram, proper trapezoid and parallelogram, proper trapezoid, parallelogram, and trapezoid...

In a graph G , a spanning tree T is said to be a tree t-spanner of the graph G if the distance between any two vertices in T is at most t times their distance in G . The tree t-spanner has many applications in networks and distributed environments. In this paper, an algorithm is presented to find a tree 3 -spanner on trapezoid graphs in ) ( 2 n O time, where n is the number of vertices of the g...