Distance Two Surjective Labelling of Paths and Interval Graphs

On Distance Two Labelling of Unit Interval Graphs

An L(2, 1)-labelling of a graph G is an assignment of non-negative integers to the vertices of G such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The L(2, 1)-labelling number of G, denoted by λ(G), is the minimum range of labels over all such labellings. In this paper, we first discuss some necessary and sufficient ...

Powers of Hamiltonian paths in interval graphs

We give a simple proof that the obvious necessary conditions for a graph to contain the kth power of a Hamiltonian path are sufficient for the class of interval graphs. The proof is based on showing that a greedy algorithm tests for the existence of Hamiltonian path powers in interval graphs. We will also discuss covers by powers of paths and analogues of the Hamiltonian completion number. c © ...

Long cycles and paths in distance graphs

For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of ...

M1 internship report: problems on interval graphs L(2,1)-labelling Almost-mixed unit interval graphs

Distance-Preserving Subgraphs of Interval Graphs

We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with k terminal vertices. We prove the following results. 1. Finding an optimal distance-preserving subgraph is NP-hard for general graphs. 2. Every interval graph admits a subgraph with O(k) branching vertices that approximates pairwise terminal distances up to an additive term of +...