Abstract
In this paper we investigate the problem of computing two parameters of a graph: the Total Interval Number TIN( G), and the Hamiltonian Completion Number HCN( G). We show that in case of a triangle-free graph these two parameters (TIN( G) and HCN( L( G)), where L( G) is the line graph of G) are related by a simple formula. We use this relationship to find a polynomial algorithm which determines TIN( G), for a tree G. Then we describe a construction which gives an interval representation of a tree G with TIN( G) number of intervals.
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