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Universal minimal total dominating functions of trees

✍ Scribed by Alan Stacey

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
179 KB
Volume
140
Category
Article
ISSN
0012-365X

No coin nor oath required. For personal study only.

✦ Synopsis

We show that any tree that has a universal minimal total dominating function has one which only takes 0-1 values. K 3 demonstrates that this fails for graphs in general.

Given a graph G =(V, E), for each vertex ve V let F(v) be the set of its neighbours (in particular, not including v itself). A total dominating function of G is a function f: V~[0, 1] such that for each v~ V, fly] := ~ f(u)>~l. u~F (v)

πŸ“œ SIMILAR VOLUMES

A characterisation of universal minimal
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✍ E.J. Cockayne; C.M. Mynhardt πŸ“‚ Article πŸ“… 1995 πŸ› Elsevier Science 🌐 English βš– 445 KB

A total dominating function (TDF) of a graph G = (V, E) is a function f: V~ [0, 1] such that for each v~ V, ~u~Ntv)f(u)>~ 1, where N(v) denotes the set of neighbours of v. Although convex combinations of TDFs are also TDFs, convex combinations of minimal TDFs (MTDFs) are not necessarily minimal. An

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The relation Ye on the set of minimal dominating functions (MDFs) of a finite graph G is defined by f&?g if and only if any convex combination off and g is also an MDF. If fis a nonintegral MDF of a tree, the existence of another MDF with fewer nonintegral values and other desirable properties is es

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## Abstract A total dominating function (TDF) of a graph __G__ = (__V, E__) is a function __f__: __V__ ← [0, 1] such that for each __v__ Ο΅ V, Ξ£~uΟ΅N(v)~ f(u) β‰₯ 1 (where __N__(__v__) denotes the set of neighbors of vertex __v__). Convex combinations of TDFs are also TDFs. However, convex combinations

Convexity of minimal total dominating fu
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A total dominating function (TDF) of a graph G = (V, E) is a function f : V β†’ [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