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More on linear and metric tree maps

Kozerenko Sergiy

Opuscula Mathematica

2021

Vol. 41, no. 1

55--70

We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.

Cryptography with right-angled Artin groups

Flores R., Kahrobaei D.

Theoretical and Applied Informatics

2016

Vol. 28, No. 3

8--16

In this paper we propose right-angled Artin groups as a platform for secret sharing schemes based on the efficiency (linear time) of the word problem. Inspired by previous work of Grigoriev-Shpilrain in the context of graphs, we define two new problems: Subgroup Isomorphism Problem and Group Homomorphism Problem. Based on them, we also propose two new authentication schemes. For right-angled Artin groups, the Group Homomorphism and Graph Homomorphism problems are equivalent, and the later is known to be NP-complete. In the case of the Subgroup Isomorphism problem, we bring some results due to Bridson who shows there are right-angled Artin groups in which this problem is unsolvable.