On the Complexity of Finding Large Odd Induced Subgraphs and Odd Colorings

摘要

We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition V(G). We call these parameters (G) and chi(odd)(G), respectively. We prove that deciding whether chi(odd)(G) = q is polynomial-time solvable if q = 2, and NP-complete otherwise. We provide algorithms in time 2O(rw) center dot nO(1) and 2O(q center dot rw) center dot nO(1) to compute chi(odd)(G) and to decide whether chi(odd)(G) = q on n-vertex graphs of rank-width at most rw, respectively, and we prove that the dependency on rank-width is asymptotically optimal under the ETH. Finally, we give some tight bounds for these parameters on restricted graph classes or in relation to other parameters.