The k th power of a graph G=(V, E), G k, is the graph whose vertex set is V and in which two
distinct vertices are adjacent if and only if their distance in G is at most k. This article proves …

The k-independence number of a graph G is the maximum size of a set of vertices at
pairwise distance greater than k. In this paper, for each positive integer k, we prove sharp …

For k≥ 1, the k-independence number α k of a graph is the maximum number of vertices that
are mutually at distance greater than k. The well-known inertia and ratio bounds for the (1-) …

We present a simple, yet efficient, heuristic for finding a large 2-independent set of regular
graphs. We analyse the average-case performance of this heuristic, which is a randomised …

While efficient algorithms for finding minimal distance-k dominating sets exist, finding
minimum such sets is NP-hard even for bipartite graphs. This paper presents a distributed …

The $ k $-independence number of a graph, $\alpha_k (G) $, is the maximum size of a set of
vertices at pairwise distance greater than $ k $, or alternatively, the independence number of …

A 3-star is a complete bipartite graph K 1, 3. A 3-star packing of a graph G is a collection of
vertex-disjoint subgraphs of G in which each subgraph is a 3-star. The maximum 3-star …

For a fixed positive integer d≥ 2, a distance-d independent set (DdIS) of a graph is a vertex-
subset whose distance between any two members is at least d. Imagine that there is a token …

In this paper we consider the problem of finding large collections of vertices and edges
satisfying particular separation properties in random regular graphs of degree r, for each …

We prove new lower and upper bounds on the higher gonalities of finite graphs. These
bounds are generalizations of known upper and lower bounds for first gonality to higher …