Constrained Edge-Splitting Problems

Tibor Jordán

November 1999


Splitting off two edges $su, sv$ in a graph $G$ means deleting $su, sv$ and adding a new edge $uv$. Let $G=(V+s,E)$ be $k$-edge-connected in $V$ ($k\geq 2$) and let $d(s)$ be even. Lovász proved that the edges incident to $s$ can be split off in pairs in a such a way that the resulting graph on vertex set $V$ is $k$-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to meet additional requirements. We prove structural properties of the set of those pairs $u,v$ of neighbours of $s$ for which splitting off $su, sv$ destroys $k$-edge-connectivity. This leads to a new method for solving problems of this type.

By applying this method we obtain a short proof for a recent result of Nagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let $G$ and $H$ be two graphs on the same set $V+s$ of vertices and suppose that their sets of edges incident to $s$ coincide. Let $G$ ($H$) be $k$-edge-connected ($l$-edge-connected, respectively) in $V$ and let $d(s)$ be even. Then there exists a pair $su, sv$ which can be split off in both graphs preserving $k$-edge-connectivity ($l$-edge-connectivity, resp.) in $V$, provided $d(s)\geq 6$. If $k$ and $l$ are both even then such a pair always exists. Using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size

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