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Displaying similar documents to “A simple proof of Whitney's Theorem on connectivity in graphs”

The contractible subgraph of 5 -connected graphs

Chengfu Qin, Xiaofeng Guo, Weihua Yang (2013)

Czechoslovak Mathematical Journal

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An edge e of a k -connected graph G is said to be k -removable if G - e is still k -connected. A subgraph H of a k -connected graph is said to be k -contractible if its contraction results still in a k -connected graph. A k -connected graph with neither removable edge nor contractible subgraph is said to be minor minimally k -connected. In this paper, we show that there is a contractible subgraph in a 5 -connected graph which contains a vertex who is not contained in any triangles. Hence, every vertex...

Connected resolvability of graphs

Varaporn Saenpholphat, Ping Zhang (2003)

Czechoslovak Mathematical Journal

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For an ordered set W = { w 1 , w 2 , ⋯ , w k } of vertices and a vertex v in a connected graph G , the representation of v with respect to W is the k -vector r ( v | W ) = ( d ( v , w 1 ) , d ( v , w 2 ) , ⋯ , d ( v , w k ) ) , where d ( x , y ) represents the distance between the vertices x and y . The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolving set for G containing a minimum number of vertices is a basis for G . The dimension dim ( G ) is the number of vertices in a basis for G . A resolving set W of G is connected...

The all-paths transit function of a graph

Manoj Changat, Sandi Klavžar, Henry Martyn Mulder (2001)

Czechoslovak Mathematical Journal

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A transit function R on a set V is a function R V × V → 2 V satisfying the axioms u ∈ R ( u , v ) , R ( u , v ) = R ( v , u ) and R ( u , u ) = { u } , for all u , v ∈ V . The all-paths transit function of a connected graph is characterized by transit axioms.