Anja Pruchnewski

On Dominating Sets and Independent Sets of Graphs

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 ≤ ki ≤ di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ VsDk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) ∣ pi ∈ R, 0 ≤ pi ≤ 1, i = 1, …, n}. An O(Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with ∣Dk∣≤fk(p).

A Note on the Domination Number of a Bipartite Graph

Abstract. Upper bounds on the domination number of a bipartite graph are established. The same approach leads to similar results for arbitrary graphs.

On the domination number of a graph

For a finite undirected graph G on n vertices some continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the domination number γ(G) of G.

Precoloring extension for K4-minor-free graphs

Let

List Colorings of K 5 -Minor-Free Graphs With Special List Assignments

G=(V,E)

Sum choice number of generalized θ-graphs

be a graph where every vertex

( P , Q ) -Total ( r , s ) -colorings of graphs

v \in V

A planarity criterion for cubic bipartite graphs

is assigned a list of available colors

Precoloring extension for K 4 -minor-free graphs

L(v)

Weights of induced subgraphs in K 1 , r -free graphs

. We say that