Jerzy Topp

Well covered simplicial, chordal, and circular arc graphs

A graph G is called well covered if every two maximal independent sets of G have the same number of vertices. In this paper, we characterize well covered simplicial, chordal and circular arc graphs.

On packing and covering numbers of graphs

Abstract In this paper we present a characterization of connected graphs of order ( k +1) n with k -covering number n , a characterization of trees in which the k -packing and k -covering numbers are the same, and we prove that the smallest number of subgraphs of diameter at most k which cover the vertices of a block graph equals the k -packing number.

On graphs with equal domination and independent domination numbers

Abstract Allan and Laskar have shown that K 1.3 -free graphs with equal domination and independent domination numbers. In this paper new classes of graphs with equal domination independent domination numbers.are presented. In particular, the result of Allan and Laskar is generalized.

Sufficient conditions for equality of connectivity and minimum degree of a graph

For a graph G, let n(G), κ(G) and δ(G) denote the order, the connectivity, and the minimum degree of G, respectively. The paper contains some conditions on G implying κ(G) = δ(G). One of the conditions is that n(G) ≤ δ(G)(2p −1)/(2p −3) if G is a p-partite graph.

On unique independent sets in graphs

Abstract For a nonnegative integer k , a subset I of the vertex set V ( G ) of a simple graph G is said to be k -independent if I is independent and every independent subset I ′ of G with | I ′|⩾| I |−( k −1) is a subset of I . A set I of vertices is called a strong k -independent set of G if I is k -independent and the set V ( G )− I is independent in G . First we give several characterizations of k -independent sets for some classes of graphs. Then we characterize trees which have (strong) k -independent sets. Finally, we obtain lower bounds on the number of edges in graphs which have k -independent sets.

a k - and U k -stable graphs

Abstract A set I of vertices of a graph G is k-independent if the distance between every two vertices of I is at least k +1. The k-independence number, α k ( G ), is the cardinality of a maximum k-independent set of G. A set D of vertices of G is k-dominating if every vertex in V ( G )− D is at distance at most k from some vertex in D. The k-domination number, γ k ( G ), is the cardinality of a minimum k-dominating set of G. A graph G is α k -stable ( γ k -stable) if α k ( G − e )= α k ( G ) ( γ k ( G − e )= γ k ( G )) for every edge e of G. We establish conditions under which a graph is α k - and γ k -stable. In particular, we give constructive characterizations of α k - and γ k -stable trees.

On Domination and Independence Numbers of Graphs

abstractA set S of vertices of a graph G is dominating if each vertex x not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number, γ(G), is the order of the smallest dominating set in G. The independence number, α(G), is the order of the largest independent set in G. In this paper we characterize bipartite graphs and block graphs G for which γ(G) = α(G).

Graphs with unique minimum edge dominating sets and graphs with unique maximum independent sets of vertices

Abstract A set I of vertices of a graph G is an independent set if no two vertices of I are adjacent. A set M of edges of G is an edge dominating set if each edge not in M is adjacent to at least one edge in M . We investigate graphs that have unique minimum edge dominating sets. Moreover, we characterize graphs whose total graphs (line graphs) have unique maximum independent sets of vertices.

Totally equimatchable graphs

Abstract A subset X of vertices and edges of a graph G is totally matching if no two elements of X are adjacent or incident. In this paper we determine all graphs in which every maximal total matching is maximum.

Interpolation theorems for domination numbers of a graph

An integer-valued graph function π is an interpolating function if for every connected graph G, π(T(G)) is a set of consecutive integers, where T(G) is the set of all spanning trees of G. The interpolating character of a number of domination related parameters is considered.