Discussiones Mathematicae Graph Theory 33(1) (2013) 71-89
doi: 10.7151/dmgt.1653

Dedicated to Mietek Borowiecki on the occasion of his 70th birthday.

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On Vertices Enforcing a Hamiltonian Cycle

Igor Fabrici

Institute of Mathematics
P.J. Šafárik University in Košice, Slovak Republic

Erhard Hexel

Institut für Mathematik
Technische Universität Ilmenau, Germany

Stanislav Jendrol'

Institut of Mathematics
P.J. Šafárik University in Košice, Slovak Republic

Abstract

A nonempty vertex set X ⊆ V(G) of a hamiltonian graph G is called an of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.

Keywords: cycle, hamiltonian, 1-hamiltonian

2010 Mathematics Subject Classification: 05C45.

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Received 19 April 2012
Accepted 25 October 2012