Joining caterpillars and stability of the tree center Original Research Article

The path-table image of a tree T collects information regarding the paths in T: for each vertex image, the row of image relative to image lists the number of paths containing image of the various lengths. We call this row the path-row of image in T. Two trees having the same path-table (up to reordering the rows) are called path-congruent (or path-isomorphic). Motivated by Kelly–Ulamʹs Reconstruction Conjecture and its variants, we have looked for new necessary and sufficient conditions for isomorphisms between two trees. Path-congruent trees need not be isomorphic, although they are similar in some respects. In [P. Dulio, V. Pannone, Trees with path-stable center, Ars Combinatoria, LXXX (2006) 153–175] we have introduced the concepts of trunk image of a tree T and ramification image of a vertex image, and proved that, if the ramification of the central vertices attains its minimum or maximum value, then the path-row of a central vertex is “unique”, i.e. it is different from the path-row of any non-central vertex (in fact, this uniqueness property of a central path-row holds for all trees of diameter less than 8, regardless of the ramification values). In this paper we prove that, for all other values of the ramification, and for all diameters greater than 7, there are trees in which the above uniqueness fails.