Basic embeddings into a product of graphs
If \(K\), \(X\), \(Y\) are topological spaces, then an embedding \(K\subset X\times Y\) is called basic if for every continuous real function \(f:K\to \mathbb{R}\) there exist continuous real functions \(g:X\to \mathbb{R}\) and \(h:Y\to \mathbb{R}\) such that \(f(x,y)= g(x)+ h(y)\) for any point \((x,y)\in K\). Now, let us specify: \(K\) is a finite graph, \(X= \mathbb{R}\) and \(Y\) is a star with \(n\geq 3\) rays (or \(n\)-od) denoted by \(T_n\). A vertex \(A\) of \(K\) is called ``bad'' if either its degree is greater than 4, or its degree equals 4 and \(A\) has no hanging edges. Then the defect of \(K\) is the following sum \[ \delta(K)= (\deg A_1- 2)+\cdots+ (\deg A_k-2), \] where \(A_1,\dots, A_k\) are bad vertices of \(K\). The following theorem characterizes basically embeddable finite graphs into \(\mathbb{R}\times T_n\): A finite graph \(K\) is basically embeddable into \(\mathbb{R}\times T_n\) if and only if \(K\) is a tree and either \(\delta(K)< n\) or \(\delta(K)= n\) and \(K\) has a bad vertex with hanging edge. It is shown as a corollary that there exist only a finite number of ``prohibited'' subgraphs for basic embeddings into \(\mathbb{R}\times T_n\), therefore, for a finite graph \(K\) there is an algorithm for checking whether \(K\) is basically embeddable into \(\mathbb{R}\times T_n\). The paper includes results related to the ambient space \(T_m\times T_n\) \((m\geq 3)\).
scientific article; zbMATH DE number 1421582