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 pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solvin...

Hierarchical structures and uncertainty measures are two main aspects in granular computing, approximate reasoning and cognitive process. Typical hesitant fuzzy sets, as a prime extension of fuzzy sets, are more flexible to reflect the hesitance and ambiguity in knowledge representation and decision making. In this paper, we mainly investigate the hierarchical structures and uncertainty measure...

The Zarankiewicz number z(b; s) is the maximum size of a subgraph of Kb,b which does not contain Ks,s as a subgraph. The two-color bipartite Ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of Kb,b with two colors contains a Ks,s in the rst color or a Kt,t in the second color.In this work, we design and exploit a computational method for bounding and computing...

A generic de.nition of fuzzy morphism between graphs (GFM) is introduced that includes classical graph related problem de.nitions as sub-cases (such as graph and subgraph isomorphism). The GFM uses a pair of fuzzy relations, one on the vertices and one on the edges. Each relation is a mapping between the elements of two graphs. These two fuzzy relations are linked with constraints derived from ...

Characterising graph classes by forbidding a set of graphs has been a common feature in graph theory for many years. Up to date many classes have been char-acterised with respect to the induced subgraph relation and there are a plethora of results regarding graph classes closed under the minor relation. We consider the characterisation of graph classes closed under partial orders including edge...

Perfect vertex elimination schemes are part of the characterizations for several classes of graphs, including chordal and cop-win. Partial elimination schemes reduce a graph to an important subgraph, for example, k-cores and robber-win graphs. We are interested in those partial elimination schemes, in which once a vertex can be eliminated it is always ready to be eliminated. In such a scheme, t...

This paper discusses the game colouring number of partial k-trees and planar graphs. Let colg(PT k) and colg(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this paper, we prove that colg(PT k) = 3k + 2 and colg(P) ≥ 11. We also prove that the game colouring number colg(G) of a graph is a monotone parameter,...

In this paper we begin the classification completed in [12] of all partial linear spaces n , graphs F, and groups G which satisfy one of the following: I. II = (0>, ££) is a connected partial linear space of order 2 in which every pair of intersecting lines lies in a subspace isomorphic to the dual of an affine plane of order 2; II. F is a connected graph such that, for each vertex x of F, the ...

To use fuzzy controllers for automization tasks appropriate fuzzy sets and fuzzy rules have to be deened. This can be diicult in some domains, and the resulting controller has to be tuned. Neuro{fuzzy models can help in this tuning process by adapting fuzzy sets and creating fuzzy rules. Combinations of neural networks and fuzzy controllers are suitable if there is only partial knowledge in the...

Let G be a graph and W a subset of V (G). Let g, f : V (G) → Z be two integer-valued functions such that g(x) ≤ f(x) for all x ∈ V (G) and g(y) ≡ f(y) (mod 2) for all y ∈ W . Then a spanning subgraph F of G is called a partial parity (g, f)-factor with respect to W if g(x) ≤ degF (x) ≤ f(x) for all x ∈ V (G) and degF (y) ≡ f(y) (mod 2) for all y ∈ W . We obtain a criterion for a graph G to have...