## Complexity of two coloring problems in cubic planar bipartite mixed graphs

- Bernard Ries
- Discrete Applied Mathematics
- 2010

3 Excerpts

- Published 2007 in PPAM

We consider the mixed graph coloring problem. A mixed graph GM = (V,E,A) is a graph with vertex set V and containing edges (set E) and arcs (set A). An edge joining vertices i and j is denoted by {i, j}, while an arc with tail p and head q is denoted by (p, q). A kcoloring of GM is a function φ : V → {1, 2, . . . , k} such that φ(i) 6= φ(j) for {i, j} ∈ E and φ(p) < φ(q) for (p, q) ∈ A. Observe that the mixed graph GM must be acyclic, i.e., must not contain any directed circuit, otherwise no proper k-coloring exists. The mixed graph coloring model can be used for formulating scheduling problems where both incompatibility and precedence constraints are present. Formally, let T be a collection of jobs (with unit processing times). These jobs have to be processed taking into account the following constraints: 1. Precedence constraints. There is a set of ordered jobs (i, j) such that i must be processed before j. 2. Disjunctive constraints. For a family I = {I1, . . . , Il} of subsets of T , no two jobs in Iα can be processed simultaneously, α = 1, . . . , l. Consider now a mixed graph GM = (V,E,A) obtained as follows: 1. With each job j in T we associate a vertex j in V . GM currently has no other vertices, and no arcs and edges. 2. For each ordered pair (i, j) of jobs we introduce an arc (i, j) in GM . 3. For each subset Iα, α = 1, . . . , l, we introduce a clique associated with the jobs in Iα. (If an edge is needed between vertices i and j, we introduce it only if there was no previous arc or edge joining i and j.) It easy to see that there is one-to-one correspondence between feasible schedules in k time units and k-colorings of the mixed graph GM . Herein, we present an O(n log n) time algorithm for finding the optimal scheduling in the case when incompatibility and precedence constraints form a series-parallel mixed graph.

@inproceedings{Furmanczyk2007SchedulingWP,
title={Scheduling with Precedence Constraints: Mixed Graph Coloring in Series-Parallel Graphs},
author={Hanna Furmanczyk and Adrian Kosowski and Pawel Zylinski},
booktitle={PPAM},
year={2007}
}