Complexity: an approach through independence

The complexity of an edge of a clutter is the ratio of the size of its minimum subset, that is present only in this given edge, to the size of the edge. The complexity of a clutter is the maximum of the complexities of its edges. We study the complexity of clutters arising from independent sets and matchings of graphs. The graphs considered in this paper are finite, undirected and do not contain multiple edges or loops. For a graph G let V (G) and E(G) denote the sets of vertices and edges of G, respectively. For a vertex v ∈ V (G) let d(v) denote the degree of v, and let ∆(G) be the maximum degree of a vertex of G. If E ⊆ E(G) then let V (E) be the set of vertices of G which are incident to an edge from E. For S ⊆ V (G) let G[S] denote the subgraph of G induced by the set S. If u, v are vertices of a graph G, then let ρ(u, v) denote the distance between the two vertices, and let diam(G) denote the diameter of G. For a positive integer n let K n denote the complete graph on n vertices. If m and n are positive integers then assume K m,n to be the complete bipartite graph one side of which has m vertices and the other side n vertices. We also consider clutters. A clutter L, is a pair (V, E), where V is a finite set and E is a family of subsets of V none of which is a subset of another. Following [ 2], the elements of V will be called vertices of L, while the elements of E-edges of L. If for a graph G = (V, E) we denote the set of all maximal independent sets of the graph G by U G , then (V, U G) is a clutter. In our paper we identify the set U G and the clutter (V, U G), and use the same notation U G for both of them.