Changing upper irredundance by edge addition

AbstractDenote the upper irredundance number of a graph G by IR(G). A graph G is IR-edge-addition-sensitive if its upper irredundance number changes whenever an edge of Ḡ is added to G. Specifically, G is IR-edge-critical (IR+-edge-critical, respectively) if IR(G+e)<IR(G) (IR(G+e)>IR(G), respectively) for each edge e of Ḡ. We show that if G is IR-edge-addition-sensitive, then G is either IR-edge-critical or IR+-edge-critical. We obtain properties of the latter class of graphs, particularly in the case where β(G)=IR(G)=2 (where β(G) denotes the vertex independence number of G). This leads to an infinite class of IR+-edge-critical graphs where IR(G)=2

Balancing equal weights on the integer line

AbstractThe problem is to determine the function f(n, r) which is the total number of balance positions when r equal weights are placed on a centrally pivoted uniform rod at r distinct points whose coordinates with respect to the center as origin are a subset of the 2n+1 integers {0, ±1, ±2, …, ±n}. The function is evaluated precisely for small n and estimated accurately for all n using asymptotic statistical theory

On generalised minimal domination parameters for paths

AbstractA subset X of vertices of a graph is a k-minimal P-set if X has property P, but the removal of any l vertices from X, where l⩽k, followed by the addition of any(l−1) vertices destroys the property P. We note that 1-minimality is the usual minimality concept. In this paper we determine Γk(Pn), the largest cardinality of a k-minimal dominating set of the n-vertex path Pn. We also prove for any n-vertex graph G, Γ2(G)γ(Ḡ)⩽n and finally a ‘Gallai-type’ theorem for k-minimal parameters is established

Broadcasts and domination in trees

A broadcast on a graph G is a function f:V→Z+∪0. The broadcast number of G is the minimum value of ∑v∈Vf(v) among all broadcasts f for which each vertex of G is within distance f(v) from some vertex v with f(v

On i−-ER-critical graphs

AbstractWe are interested in the behaviour of the independent domination number i(G) of a graph G under edge deletion, and in particular in i−-ER-critical graphs, i.e., graphs for which i(G) decreases whenever an edge e is removed. If γ(G) denotes the domination number of G, we determine all the i−-ER-critical graphs G such that γ(G)=2 and i(G−e)=2 for every edge e of G. Different classes of i−-ER-critical graphs such that i(G−e)>γ(G) for all or some edges e are described. Finally, for a particular family of circulants, we find the exact value of i(G−e) for every edge e of the graphs of this family and obtain as a corollary the number of automorphism classes of their edge sets