Abstract
Let G = (V,E) be a graph. A function f : E ? [0, 1] is called an edge dominating function if ?x?N[e] f(x)?1 for all e ? E(G), where N[e] is the closed neighbourhood of the edge e. An edge dominating function f is called minimal (MEDF) if for all functions g : E ? [0,1] with g < f, g is not an edge dominating function. The fractional edge domination number ?'f and the upper fractional edge domination number ?'f are defined by ?'f (G) = min{|f| : f is an MEDF of G} and ?'f (G) = max{|f| : f is an MEDF of G}, where |f| = ?e?E f(e). Further we introduce the fractional parameters corresponding to edge irredundance and edge independence, leading to the fractional edge domination chain. We also consider topological properties of the set of all edge dominating functions of G.
Highlights
INTRODUCTIONThe minimum (maximum) cardinality of a minimal dominating set of G is called the domination number (upper domination number) of G and is denoted by γ(G) Γ(G)
By a graph G = (V, E) we mean a finite, undirected graph with neither loops nor multiple edges
Further we introduce the fractional parameters corresponding to edge irredundance and edge independence, leading to the fractional edge domination chain
Summary
The minimum (maximum) cardinality of a minimal dominating set of G is called the domination number (upper domination number) of G and is denoted by γ(G) Γ(G). A subset X of E is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The maximum cardinality of a 2-edge packing in G is called the 2-edge packing number of G and is denoted by P2(G). Cockayne and Mynhardt [8] have indicated that edge subsets may be embedded into sets of functions and an analogous concept of convexity could be developed. The maximum cardinality of a 2-packing in G is called the 2-packing number of G and is denoted by P2(G).
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