Showing papers on "K-tree published in 2021"
TL;DR: It is proved that if G is a k-tree, then there is no asymptotically better algorithm than `wait until $\frac{1}{k+1}$ fraction of vertices'.
Abstract: We study a new optimal stopping problem: Let $G$ be a fixed graph with $n$ vertices which become active on-line in time, one by another, in a random order. The active part of $G$ is the subgraph induced by the active vertices. Find a stopping algorithm that maximizes the expected number of connected components of the active part of $G$. We prove that if $G$ is a $k$-tree, then there is no asymptotically better algorithm than `wait until $\frac{1}{k+1}$ fraction of vertices'. The maximum expected number of connected components equals to $$\left(\frac{k^k}{(k+1)^{k+1}}+o(1)\right)n.$$
3 citations
TL;DR: In this article, the authors study and characterize by forbidden subposets the k-tree posets that admit a containment model mapping vertices into paths of a tree, and characterize the dually-CPT and strong-CPTs and their comparability graphs.
Abstract: A k-tree is either a complete graph on k vertices or a graph that contains a vertex whose neighborhood induces a complete graph on k vertices and whose removal results in a k-tree. If the comparability graph of a poset P is a k-tree, we say that P is a k-tree poset. In the present work, we study and characterize by forbidden subposets the k-tree posets that admit a containment model mapping vertices into paths of a tree (CPT k-tree posets). Furthermore, we characterize the dually-CPT and strong-CPT k-tree posets and their comparability graphs. The characterizations lead to efficient recognition algorithms for the respective classes.
2 citations
TL;DR: It is shown that every connected K1,k+1-free graph G has a spanning k-tree if the degree sum of any 3k − 3 independent vertices in G is at least |G | − 2, where |G| is the order of G.
Abstract: Abstract For an integer k ≥ 2, a k-tree T is defined as a tree with maximum degree at most k. If a k-tree T spans a graph G, then T is called a spanning k-tree of G. Since a spanning 2-tree is a Hamiltonian path, a spanning k-tree is an extended concept of a Hamiltonian path. The first result, implying the existence of k-trees in star-free graphs, was by Caro, Krasikov, and Roditty in 1985, and independently, Jackson and Wormald in 1990, who proved that for any integer k with k ≥ 3, every connected K1,k-free graph contains a spanning k-tree. In this paper, we focus on a sharp condition that guarantees the existence of a spanning k-tree in K1,k+1-free graphs. In particular, we show that every connected K1,k+1-free graph G has a spanning k-tree if the degree sum of any 3k − 3 independent vertices in G is at least |G| − 2, where |G| is the order of G.