Generalized vertex-rankings of trees

An optimal parallel algorithm for c-vertex-ranking of trees

“…A vertex-labeling ϕ of a tree T is a c-vertex-ranking if and only if no more than c vertices of the same rank are visible from each vertex of T under ϕ [14]. The visible-list (list for short) L(ϕ) of a c-vertex-ranking ϕ is defined as follows [5,14]:…”

“…Every critical c-vertexranking ϕ is optimal [14]. We find a critical c-vertexranking of tree T in a bottom-up manner.…”

“…For each internal vertex u, we can construct a critical c-vertexranking of T (u) from those of the subtrees rooted at u's children. One can prove the following lemma easily from the lemmas proven in [14].…”

“…Generalization of the vertex-ranking problem was introduced in [14]. For a positive integer c, a c-vertexranking of a graph G is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels > i leaves connected components, each having at most c vertices with label i.…”

“…A c-vertex-ranking of G using r c (G) ranks is called an optimal c-vertex-ranking of G. The c-vertex-ranking problem is to find an optimal c-vertex-ranking of a given graph. Zhou et al have obtained an O(cn) time sequential algorithm to solve the c-vertex-ranking problem for trees [14]. Recently, Kashem et al have presented a polynomialtime sequential algorithm and an O(log 2 n) time parallel algorithm for partial k-trees that uses polynomial number of processors on the common CRCW PRAM model [8].…”