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Convexity of minimal total dominating functions in graphs

✍ Scribed by Yu, Bo

Publisher: John Wiley and Sons
Year: 1997
Tongue: English
Weight: 122 KB
Volume: 24
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199704)24:4<313::aid-jgt3>3.0.co;2-r

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✦ Synopsis

A total dominating function (TDF) of a graph G = (V, E) is a function f : V → [0, 1] such that for each v ∈ V , the sum of f values over the open neighbourhood of v is at least one. Zero-one valued TDFs are precisely the characteristic functions of total dominating sets of G. We study the convexity of minimal total dominating functions. A minimal total dominating function (MTDF) f is called universal if convex combinations of f and any other MTDF are minimal. Generalizing and unifying two previous major results by Cockayne, Mynhardt and Yu in the area, we give a stronger sufficiency condition for an MTDF to be universal. Moreover, we define a splitting operation on a graph G, which preserves the universality. Using the operation, we give many more classes of graphs having a universal MTDF.

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