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2-factors in claw-free graphs with locally disconnected vertices

Mingqiang An, Liming Xiong, Runli Tian (2015)

Czechoslovak Mathematical Journal

An edge of G is singular if it does not lie on any triangle of G ; otherwise, it is non-singular. A vertex u of a graph G is called locally connected if the induced subgraph G [ N ( u ) ] by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph G of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex v of degree at least 3 in G , there is a nonnegative integer s such that v lies...

2-placement of (p,q)-trees

Beata Orchel (2003)

Discussiones Mathematicae Graph Theory

Let G = (L,R;E) be a bipartite graph such that V(G) = L∪R, |L| = p and |R| = q. G is called (p,q)-tree if G is connected and |E(G)| = p+q-1. Let G = (L,R;E) and H = (L',R';E') be two (p,q)-tree. A bijection f:L ∪ R → L' ∪ R' is said to be a biplacement of G and H if f(L) = L' and f(x)f(y) ∉ E' for every edge xy of G. A biplacement of G and its copy is called 2-placement of G. A bipartite graph G is 2-placeable if G has a 2-placement. In this paper we give all (p,q)-trees which...

(H,k) stable bipartite graphs with minimum size

Aneta Dudek, Małgorzata Zwonek (2009)

Discussiones Mathematicae Graph Theory

Let us call a graph G(H;k) vertex stable if it contains a subgraph H after removing any of its k vertices. In this paper we are interested in finding the ( K n , n + 1 ; 1 ) (respectively ( K n , n ; 1 ) ) vertex stable graphs with minimum size.

(H,k) stable graphs with minimum size

Aneta Dudek, Artur Szymański, Małgorzata Zwonek (2008)

Discussiones Mathematicae Graph Theory

Let us call a G (H,k) graph vertex stable if it contains a subgraph H ever after removing any of its k vertices. By Q(H,k) we will denote the minimum size of an (H,k) vertex stable graph. In this paper, we are interested in finding Q(₃,k), Q(₄,k), Q ( K 1 , p , k ) and Q(Kₛ,k).

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