TY - JOUR

T1 - Minimal k-Connected Graphs with Minimal Number of Vertices of Degree k

AU - Karpov, D. V.

N1 - Karpov, D.V. Minimal k-Connected Graphs with Minimal Number of Vertices of Degree k . J Math Sci 212, 666–682 (2016). https://doi.org/10.1007/s10958-016-2697-1

PY - 2016

Y1 - 2016

N2 - A graph is k-connected if it has at least k+1 vertices and remains connected after deleting any k−1 vertices. A k-connected graph is said to be minimal if any its subgraph obtained by deleting any edge is not k-connected. W. Mader proved that any minimal k-connected graph with n vertices has at least (Formula presented.) vertices of degree k. The main result of the present paper is that any minimal k-connected graph with minimal number of vertices of degree k is isomorphic to a graph Gk,T, where T is a tree the maximal vertex degree of which is at most k + 1. The graph Gk,Tis constructed from k disjoint copies of the tree T in the following way. If a is a vertex of T of degree j and a1,.. , akare the corresponding vertices of the copies of T, then k + 1 − j new vertices of degree k, which are adjacent to {a1,.. , ak}, are added. Bibliography: 10 titles.

AB - A graph is k-connected if it has at least k+1 vertices and remains connected after deleting any k−1 vertices. A k-connected graph is said to be minimal if any its subgraph obtained by deleting any edge is not k-connected. W. Mader proved that any minimal k-connected graph with n vertices has at least (Formula presented.) vertices of degree k. The main result of the present paper is that any minimal k-connected graph with minimal number of vertices of degree k is isomorphic to a graph Gk,T, where T is a tree the maximal vertex degree of which is at most k + 1. The graph Gk,Tis constructed from k disjoint copies of the tree T in the following way. If a is a vertex of T of degree j and a1,.. , akare the corresponding vertices of the copies of T, then k + 1 − j new vertices of degree k, which are adjacent to {a1,.. , ak}, are added. Bibliography: 10 titles.

KW - Complete Bipartite Graph

KW - Simple Path

KW - Internal Vertex

KW - Common Edge

KW - Pendant Vertex

UR - http://www.scopus.com/inward/record.url?scp=84953382210&partnerID=8YFLogxK

U2 - 10.1007/s10958-016-2697-1

DO - 10.1007/s10958-016-2697-1

M3 - Article

AN - SCOPUS:84953382210

VL - 212

SP - 666

EP - 682

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -