Packing and Covering Odd (u,v)-trails in a Graph
نویسنده
چکیده
In this thesis, we investigate the problem of packing and covering odd (u, v)-trails in a graph. A (u, v)-trail is a (u, v)-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Given a graph G and two specified vertices u and v, the odd (u, v)-trail packing number, denoted by ν(u, v), is the maximum number of edge-disjoint odd (u, v)-trails in G. And, the odd (u, v)-trail covering number, denoted by τ(u, v), is the minimum size of an edge-set that intersects every odd (u, v)-trail in G. In 2016, Churchley, Mohar, and Wu, were the first ones to prove a constant factor bound on the covering-vs.-packing ratio, by showing that τ(u, v) ≤ 8 · ν(u, v). Our main result in this thesis is an improved bound on the covering number: τ(u, v) ≤ 5 · ν(u, v) + 2. The proof leads to a polynomial-time algorithm to find, for any given k ≥ 1, either k edge-disjoint odd (u, v)-trails in G or a set of at most 5k − 3 edges intersecting all odd (u, v)-trails in G.
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Min-Max Theorems for Packing and Covering Odd (u, v)-trails
We investigate the problem of packing and covering odd (u, v)-trails in a graph. A (u, v)-trail is a (u, v)-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let ν(u, v) denote the maximum number of edge-disjoint odd (u, v)-trails, and τ(u, v) denote the minimum size of an edge-set that intersects every odd (u, ...
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