arXiv subjects

Aida Roshany-Tabrizi

Publications and source records attributed to Aida Roshany-Tabrizi.

3 recordsLinked to original sources

arXiv2025

Polynomial-time algorithms for PATH COVER and PATH PARTITION on trees and graphs of bounded treewidth

In the PATH COVER problem, one asks to cover the vertices of a graph using the smallest possible number of (not necessarily disjoint) paths. While the variant where the paths need to be pairwise vertex-disjoint, which we call PATH PARTITION, is extensively studied, surprisingly little is known about PATH COVER. We start filling this gap by designing a linear-time algorithm for PATH COVER on trees. We show that PATH COVER can be solved in polynomial time on graphs of bounded treewidth using a dynamic programming scheme. It runs in XP time $n^{t^{O(t)}}$ (where $n$ is the number of vertices and $t$ the treewidth of the input graph) or $κ^{t^{O(t)}}n$ if there is an upper-bound $κ$ on the solution size. A similar algorithm gives an FPT $2^{O(t\log t)}n$ algorithm for PATH PARTITION, which can be improved to (randomized) $2^{O(t)}n$ using the Cut\&Count technique. These results also apply to the variants where the paths are required to be induced (i.e. chordless) and/or edge-disjoint.

cs.DS↗

arXiv2025

Approximating Maximum Edge 2-Coloring by Normalizing Graphs

In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G with the goal to maximize the number of colors. For a relevant graph class, ME2C models anti-Ramsey numbers and it was considered in network applications. For the problem a 2-approximation algorithm is known, and if the input graph has a perfect matching, the same algorithm has been shown to have a performance guarantee of 5/3. It is known that ME2C is APX-hard and that it is UG-hard to obtain an approximation ratio better than 1.5. We show that if the input graph has a perfect matching, there is a polynomial time 1.625-approximation and if the graph is claw-free or if the maximum degree of the input graph is at most three (i.e., the graph is subcubic), there is a polynomial time 1.5-approximation algorithm for ME2C

cs.DM↗

arXiv2021

Characterizing extremal graphs for open neighbourhood location-domination

An open neighbourhood locating-dominating set is a set $S$ of vertices of a graph $G$ such that each vertex of $G$ has a neighbour in $S$, and for any two vertices $u,v$ of $G$, there is at least one vertex in $S$ that is a neighbour of exactly one of $u$ and $v$. We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of vertices. More precisely, we prove that these graphs are exactly the graphs all whose connected components are half-graphs (a half-graph is a special bipartite graph with both parts of the same size, where each part can be ordered so that the open neighbourhoods of consecutive vertices differ by exactly one vertex). This corrects a wrong characterization from the literature.

math.CO↗