Michael S. Jacobson

The bondage number of a graph

Abstract A set D of vertices in a graph G is a dominating set if each vertex of G that is not in D is adjacent to at least one vertex of D. The minimum cardinality among all dominating sets in G is called the domination number of G and denoted σ(G). We define the bondage number b(G) of a graph G to be the cardinality of a smallest set E of edges for which σ(G−E)>σ(G). Sharp bounds are obtained for b(G), and the exact values are determined for several classes of graphs.

On graphs having domination number half their order

In this paper we present a characterization of connected graphs of order 2n with domination numbern. Using this class of graphs, we determine an infinite class of graphs with the property that the domination number of the product of any two is precisely the product of the domination numbers.

Neighborhood unions and hamiltonian properties in graphs

We investigate the relationship between the cardinality of the union of the neighborhoods of an arbitrary pair of nonadjacent vertices and various hamiltonian type properties in graphs. In particular, we show that if G is 2-connected, of order p ≥ 3 and if for every pair of nonadjacent vertices x and y: 1. (a) ∥N(x) ⌣ N(y)∥ ≧ (p − 1)2, then G is traceable, 2. (b) ∥N(x) ⌣ N(y)∥ ≧ (2p − 1)3, then G is hamiltonian, and if G is 3-connected and 3. (c) ∥N(x) ⌣ N(y)∥ ≧ 2p3, then G is hamiltonian-connected.

Chordal graphs and upper irredundance, upper domination and independence

Abstract In this paper we consider the following parameters: IR( G ), the upper irredundance number, which is the order of the largest maximal irredundant set, Γ ( G ), the upper domination number, which is the order of the largest minimal dominating set and β ( G ), the independence number, which is the order of the largest maximal independent set. It is well known that for any graph G , β(G) ⩽ Γ(G) ⩽ IR (G) . In this paper we show that these parameters are equal for all chordal graphs, and a class of graphs not containing a set of forbidden subgraphs.

Forbidden subgraphs and Hamiltonian properties of graphs

Various sufficient conditions are given, in terms of forbidden subgraphs, that imply a graph is either homogeneously traceable, hamiltonian or pancyclic.

Irregular networks, regular graphs and integer matrices with distinct row and column sums

Abstract A network is a simple graph to which each edge has been assigned a positive integer weight. A network is irregular if the sum of the edges incident to each vertex is distinct. In this paper we study this concept for regular or nearly regular graphs and derive a relationship to integer matrices with distinct row and column sums. In particular, we consider the parameter, s(G), the irregularity strength of a graph G, which is the smallest maximum weight over all irregular networks with underlying graph G. It is known that if G is an r-regular graph of order n, then s(G)⩾(n+r−1) r . We exhibit infinitely many r-regular graphs with s(G)=[(n+r−1) r] , and it is proved that s(G)⩽ [ n 2 ]+2 , for all r-regular graphs on n vertices if r is even. We also study totally irregular matrices, that is positive integer matrices with distinct row and column sums having the smallest possible maximal entry. As a corollary, we can determine the strength of complete bipartite graphs Kp,q except in the case when p=q is odd.

On k-ordered graphs

We prove a hypergraph version of Halls theorem. The proof is topological.

On the domination of the products of graphs II: Trees

For a graph G, a subset of vertices D is a dominating set if for each vertex X not in D, X is adjacent to at least one vertex of D. The domination number, γ(G), is the order of the smallest such set. An outstanding conjecture in the theory of domination is for any two graph G and H, One result presented in this paper settles this question in the case when at least one of G or H is a tree. We show that for all graphs G and any tree T. Furthermore, we supply a partial characterization for which pairs of trees, T1 and T2, strict inequality occurs. We show for almost all pairs of trees.

General results on tolerance intersection graphs

In this paper, general results are presented that are related to ϕ-tolerance intersection graphs previously defined by Jacobson, McMorris, and Mulder. For example, it is shown that all graphs are ϕ-tolerance intersection graphs for all ϕ, yet for “nice” ϕ, almost no graphs are ϕ-tolerance interval graphs. Additional results about representation of trees are given.

Packing of graphic n -tuples

An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v1, …, vn} in which the degree of vi is the ith entry of π. Graphic n-tuples (d, …, d) and (d, …, d) pack if there are edge-disjoint n-vertex graphs G1 and G2 such that d(vi) = d and d(vi) = d for all i. We prove that graphic n-tuples π1 and π2 pack if , where Δand δdenote the largest and smallest entries in π1 + π2 (strict inequality when δ = 1); also, the bound is sharp. Kundu and Lovasz independently proved that a graphic n-tuple π is realized by a graph with a k-factor if the n-tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge-disjoint 1-factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k⩽3.