Graphs with homeomorphically irreducible spanning trees

Summary

By a graph G we mean here a linear network, at least 3-connected with no vertices of degree 2, no multiple edges and no loops. A graph T is called a spanning graph of G provided (i) T is a tree, (ii) T is a subgraph of G and (iii) every vertex of G belongs to T.

A graph is said to be homeomorphically irreducible (HI) if it has no nodes of degree 2. Thus an HI tree may be a spanning tree of G and we shall call this graph a HISTree of G.

Starting with the example of the set of all 3-polytopes, inspection shows that while all possess spanning trees [1] not all have HISTrees.⁽¹⁾ We would like to know which graphs possess HISTrees, if such graphs are common and what may be found as contingent with the existence or non-existence of this feature.

Types of HISTrees

For the purpose of the present note it is sufficient to regard these graphs as belonging to three main sub-species. These are:

1. (i) Stars (as in fig. 4, which shows the smallest).

2. (ii) Star chains (as in figs. 5, 6. 1 and 10. 1).

3. (iii) Star trees (as in fig. 7.1).

Types of 3-polytopes with HISTrees

The most obvious example of a family of graphs G containing a HISTree is the family of those 3-polytopes in which those edges not belonging to the HISTree comprise a circuit joining the terminal vertices of the HISTree.