## A characterisation of universal minimal total dominating functions in trees.(English)Zbl 0824.05034

A total dominating function (TDF) of a graph $$G= (V, E)$$ is a mapping $$f$$ of $$V$$ into the closed interval $$[0, 1]$$ of real numbers with the property that for each $$v\in V$$ the sum of values of $$f$$ in all vertices adjacent to $$v$$ is at least 1. A TDF is called minimal (MTDF), if there exists no TDF $$g$$ of $$G$$ with the property that $$g(x)\leq f(x)$$ for all $$x\in V$$ and $$g(x_ 0)< f(x_ 0)$$ at least for one $$x_ 0\in V$$. A convex combination of two functions $$f$$, $$g$$ is a function $$h_ \lambda$$ given by $$h_ \lambda(x)= \lambda f(x)+ (1- \lambda) g(x)$$, where $$\lambda$$ is a real number from the interval $$[0,1]$$. If a MTDF has the property that its convex combination with any other MTDF is also a MTDF, it is called a universal MTDF. In the paper the universal minimal total dominating functions of trees are characterized.

### MSC:

 05C35 Extremal problems in graph theory 05C05 Trees 05C75 Structural characterization of families of graphs
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### References:

 [1] Cockayne, E.J.; Mynhardt, C.M.; Yu, B., Universal minimal total dominating functions in graphs, Networks, 24, 83-90, (1994) · Zbl 0804.90122 [2] E.J. Cockayne, C.M. Mynhardt and B. Yu, Total dominating functions in trees: minimality and convexity, J. Graph Theory, to appear. · Zbl 0819.05035 [3] A. Stacey, Universal minimal total dominating functions of trees, preprint. · Zbl 0824.05036 [4] Yu, B., Convexity of minimal total dominating functions in graphs, () · Zbl 0876.05048
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