Arbitrarily vertex decomposable caterpillars with four or five leaves

Abstract

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a 1 , . . . , a k ) of positive integers such that a 1 +. . .+a k = n there exists a partition (V 1 , . . . , V k ) of the vertex set of G such that for each i ∈ {1, . . . , k}, V i induces a connected subgraph of G on a i vertices.D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. We also describe two families of 292

Arbitrarily vertex decomposable suns with few rays

“…Analogous, but more intricate characterizations of avd caterpillars with more hanging vertices have been recently obtained by Cichacz et al [3].…”

“…A tree T is on-line avd if and only if either T is a path or T is a caterpillar Cat(a, b) with a and b given in Table 1 or T is the tripode S (3,5,7).…”

“…[1][2][3][4][5][6][7][8][9]). This notion originated from some applications to computer networks (cp.…”

Partitioning powers of traceable or hamiltonian graphs

“…The investigation of AP trees gained lots of attention in this context, since a connected graph is AP if one of its spanning trees is AP. It turned out, however, that the structure of AP trees is not obvious in general (see for instance [3], [4], [5] or [14]). …”

Dense Arbitrarily Vertex Decomposable Graphs

“…There are various results characterizing different families of avd trees [1,2,5,7,8]. The complete characterization of on-line avd trees was recently found by Horňák et al [9].…”