Preben Dahl Vestergaard

The number of independent sets in unicyclic graphs

In this paper, we determine upper and lower bounds for the number of independent sets in a unicyclic graph in terms of its order. This gives an upper bound for the number of independent sets in a connected graph which contains at least one cycle. We also determine the upper bound for the number of independent sets in a unicyclic graph in terms of order and girth. In each case, we characterize the extremal graphs.

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.

Connected Factors in Graphs --- a Survey

? Users may download and print one copy of any publication from the public portal for the purpose of private study or research. ? You may not further distribute the material or use it for any profit-making activity or commercial gain ? You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and investigate your claim.

Odd factors of a graph

LetG be a graph and letf be a function defined on V(G) such that f(x) is a positive odd integer for everyx ɛ V(G). A spanning subgraphF ofG is called a [l,f]-odd factor of G if dF(x) ɛ {1,3,2026, f(x)} for every x ɛV(G), wheredF(x) denotes the degree of x inF. We discuss several conditions for a graphG to have a [1,f]-odd factor.

Domination in partitioned graphs

Let V1, V2 be a partition of the vertex set in a graph G, and let γi denote the least number of vertices needed in G to dominate Vi. We prove that γ1 + γ2 ≤ 45 |V (G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ1 + γ2 for graphs with minimum valency δ, and conjecture that γ1 + γ2 ≤ 4 δ+3 |V (G)| for δ ≤ 5. As δ gets large, however, the largest possible value of (γ1 + γ2)/|V (G)| is shown to grow with the order of log δ δ .

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.

Well-covered graphs and factors

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1,2]-factor FG, i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α(G) = α(FG) for some perfect [1,2]-factor FG. This class contains all well-covered graphs G without isolated vertices of order n with α ≥ (n - 1)/2, and in particular all very well-covered graphs.

Finite and Infinite Graphs whose Spanning Trees are Pairwise Isomorphic

A theorem characterizing graphs with only one isomorphism class of spanning trees was given in [2], [3], [5], [6], and [14]. Here, two new proofs of the theorem are given, one of which has the merit that it uses only the property that an isomorphism from one tree to another maps a path of length d to a path of length d. The other proof is by induction, and it is interesting because it rapidly shows that it is sufficient to examine the unicyclic graphs with the property that every second vertex on the cycle has valency two. Two new analogues of the theorem are stated: one with labelled vertices (given in the induction proof) and one for oriented graphs. A counterexample shows that the characterization does not hold for infinite graphs, but a conjecture is stated.

On well-covered graphs of odd girth 7 or greater

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth of the graph. We prove that every isolate-vertex-free well-covered graph G containing neither C3, C5 nor C7 as a subgraph is even very well-covered. Here, a isolate-vertex-free well-covered graph G is called very well-covered, if G satisfies α(G) = n/2. A vertex set D of G is dominating if every vertex not in D is adjacent to some vertex in D. The domination number γ(G) is the minimum order of a dominating set of G. Obviously, the inequality γ(G) ≤ α(G) holds. The family Gγ=α of graphs G with γ(G) = α(G) forms a subclass of well-covered graphs. We prove that every connected member G of Gγ=α containing neither C3 nor C5 as a subgraph is a K1, C4, C7 or a corona graph.

Graphs with one isomorphism class of spanning unicyclic graphs

Abstract It is shown that a locally finite graph has exactly one isomorphism class of spanning unicyclic subgraphs if and only if either G is a unicyclic graph or G is a K2,3 with a rooted tree A attached to each vertex of one colour class and a rooted tree B attached to each vertex of the other colour class. For clarity the finite case is stated separately.