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- ArtykułDostęp ograniczonyBounds on the 2-domination number in cactus graphs(2006) Chellali, MustaphaA 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in S is dominated at least twice. The minimum cardinality of a 2-dominating set of G is the 2-domination number γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) - k(G), and if G is a graph of order n with at most one cycle, then γ2(G) ≥ (n + l - s)/2 improving Fink and Jacobson's lower bound for trees with l > s, where γt(G), l and s are the total domination number, the number of leaves and support vertices of G, respectively. We also show that if T is a tree of order n ≥ 3, then γ2(T) ≤ β(T) + s - 1, where β(T) is the independence number of T.
- ArtykułDostęp ograniczonyClassical solutions of initial problems for quasilinear partial functional differential equations of the first order(2006) Czernous, WojciechWe consider the initial problem for a quasilinear partial functional differential equation of the first order ∂tz(t, x) +Xn i=1 fi(t, x, z(t,x))∂xi z(t, x) = G(t, x, z(t,x)), z(t, x) = ϕ(t, x) ((t, x) ∈ [−h0, 0] × R n ) where z(t,x) : [−h0, 0] × [−h, h] → R is a function defined by z(t,x)(τ, ξ) = z(t + τ, x + ξ) for (τ, ξ) ∈ [−h0, 0]×[−h, h]. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.
- ArtykułDostęp ograniczonyTrees whose 2-domination subdivision number is 2(2012) Atapour, Maryam; Sheikholeslami, Seyed Mahmoud; Khodkar, AbdollahA set S of vertices in a graph G = (V,E) is a 2-dominating set if every vertex of V \S is adjacent to at least two vertices of S. The 2-domination number of a graph G, denoted by γ2(G), is the minimum size of a 2-dominating set of G. The 2-domination subdivision number sdγ2 (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-domination number. The authors have recently proved that for any tree T of order at least 3, 1 ≤ sdγ2 (T ) ≤ 2. In this paper we provide a constructive characterization of the trees whose 2-domination subdivision number is 2.