In the paper aspect of comparability alternatives in decision making problem by imprecise or incomplete information isexamined. In particular, new definitions of transitivity based on the measure of the intensity preference between pairsof alternatives in epistemic and ontic case is presented and its application to solve decision making problem is proposed.

An interval coloring of a weighted graph with non-negative weights, maps each vertex onto an open interval on the real line with width equal to the weight of the vertex, such that adjacent vertices are mapped to disjoint intervals. The total width of an interval coloring is defined as the width of the union of the intervals. The interval chromatic number of a weighted graph is the least total w...

Orthogonal ray graphs are the intersection graphs of horizontal and vertical rays (i.e. half-lines) in the plane. If the rays can have any possible orientation (left/right/up/down) then the graph is a 4-directional orthogonal ray graph (4-DORG). Otherwise, if all rays are only pointing into the positive x and y directions, the intersection graph is a 2-DORG. Similarly, for 3-DORGs, the horizont...

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing ...

Given a class of structures with a notion of connectedness (satisfying some reasonable assumptions), we consider the limit (as n -+ oc) of the probability that a random (labelled or unlabelled) n-element structure in the class is connected. The paper consists of three parts: two specific examples, N-free graphs and posets, where the limiting probability of connectedness is one-half and the gold...

Let G be an interval graph. The layout that arranges the intervals in order by right endpoint easily shows that the bandwidth of G is at most its maximum degree ∆. Hence, if G contains a clique of size ∆ + 1, then its bandwidth must be ∆. In this paper we prove a Brooks-type bound on the bandwidth of interval graphs. Namely, the bandwidth of an interval graph is at most ∆, with equality if and ...

PrCa, P., Graphs and topologies on discrete sets, Discrete Mathematics 103 (1992) 189-197. We show that a graph admits a topology on its node set which is compatible with the usual connectivity of undirected graphs if, and only if, it is a comparability graph. Then, we give a similar condition for the weak connectivity of oriented graphs and show there is no topology which is compatible with th...

In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class...

Middendorf, M. and F. Pfeiffer, Weakly transitive orientations, Hasse diagrams and string graphs, Discrete Mathematics 111 (1993) 393-400. We introduce the notion of a weakly transitive orientation for graphs as a natural generalization of transitive orientations and give a characterization for weakly transitive orientations in terms of forbidden substructurs. As a corollary of this characteriz...

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing ...