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Let G = ( V , E ) be an undirected graph and r be a vertex weight function with positive integer values. A subset (clique) D ⊆ V is an r - dominating set ( clique ) in G iff for every vertex v ϵ V there is a vertex u ϵ D with dist ( u , v ) ⩽ r ( v ). This paper contains the following...
Consider an undirected hypergraph H = ( X , E ) with a probability distribution P on the set E of its hyperedges. We investigate the average case complexity L ( H , P ) of finding an unknown hyperedge e ∗ ϵ E , chosen according to P , if only tests are allowed that check, whether e ∗ is...
The well-known concept of a polynomial function (mod m ) has been generalized to polynomial functions from Z n to Z m , and a number of results have been obtained in (Chen, 1995). In the present paper, we further define the concept of polynomial functions from Z n 1 × Z n 2 × … × Z n r to Z...
For a graph G , the path number τ ( G ) is defined as the order of a longest path in G . An ( m , k ) τ -colouring of a graph H is a partition of the vertex set of H into m subsets such that each subset induces a subgraph of H for which τ is at most k . The k − τ -chromatic number χ k τ (...
In this note we are dealing with Ulam's reconstruction conjecture applied to finite partially ordered sets. We provide a new step towards a positive answer giving a constructive proof that width two posets are reconstructible.
A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a circuit in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. A graph is called k cross-cap embeddable if it...
A hidden set is a set of points such that no two points in the set are visible to each other. In this paper we study hidden sets of points in arrangements of segments, and we provide bounds for its maximum size that are optimal up to a factor 2.
We show that the question “Is a graph 3-colorable?” remains NP-complete when restricted to the class of triangle-free graphs with maximum degree 4. Likewise the question “Is a triangle-free graph k -colorable?” is shown to be NP-complete for any fixed value of k ⩾ 4.
Bermond et al. (2) conjectured that the edge set of a cubic graph G can be partitioned into two k - linear forests , that is to say two forests whose connected components are paths of length at most k , for all k ⩾ 5. We shall prove a weaker result that the statement is valid for all k ⩾ 18.
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