Data structures and network algorithms
Data structures and network algorithms
Graph minors. V. Excluding a planar graph
Journal of Combinatorial Theory Series B
Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
A min-max relation for stable sets in graphs with no odd-K4
Journal of Combinatorial Theory Series B
Finding approximate separators and computing tree width quickly
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Stability critical graphs and even subdivisions of K4
Journal of Combinatorial Theory Series B
Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
Decision algorithms for unsplittable flow and the half-disjoint paths problem
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Highly connected sets and the excluded grid theorem
Journal of Combinatorial Theory Series B
A Polynomial Solution to the Undirected Two Paths Problem
Journal of the ACM (JACM)
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Dynamic Programming on Graphs with Bounded Treewidth
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture
Journal of Combinatorial Theory Series B
Packing Non-Zero A-Paths In Group-Labelled Graphs
Combinatorica
Non-zero disjoint cycles in highly connected group labelled graphs
Journal of Combinatorial Theory Series B
A nearly linear time algorithm for the half integral disjoint paths packing
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Some remarks on the odd hadwiger’s conjecture
Combinatorica
On the odd-minor variant of Hadwiger's conjecture
Journal of Combinatorial Theory Series B
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
Note: Note on coloring graphs without odd-Kk-minors
Journal of Combinatorial Theory Series B
Combinatorica
An improved algorithm for finding cycles through elements
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Faster approximation schemes and parameterized algorithms on H-minor-free and odd-minor-free graphs
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Shortest cycle through specified elements
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs
Theoretical Computer Science
Graph minors and parameterized algorithm design
The Multivariate Algorithmic Revolution and Beyond
Parameterized tractability of multiway cut with parity constraints
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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A totally odd K4-subdivision is a subdivision of K4 where each subdivided edge has odd length. The recognition of a totally odd K4-subdivision plays an important role in both graph theory and combinatorial optimization. Sewell and Trotter [53], Zang [63] and Thomassen [60] independently conjectured the existence of a polynomial time recognition algorithm. In this paper, we give the first polynomial time algorithm for solving this problem. We also study the the parity two disjoint rooted paths problem where we determine if there exists two vertex disjoint paths of a specified parity between two pairs of terminals. Using a similar technique, we give an O(|E(G)||V(G)|α(|E(G)|,|V(G)|)) algorithm for the parity two disjoint rooted paths problem on an input graph G, where α(|E(G)|,|V(G)|) is the inverse of the Ackermann function. We note that this clearly gives an algorithm for the well-known non-parity version of the two disjoint rooted paths problem [19, 50, 52, 55, 58]. We then extend our approach to give a polynomial time algorithm which determines, for any fixed k, whether there exists a cycle of a given parity through k independent input edges. This generalizes the non-parity version of the algorithm in [22]. Thomassen [61] gave a polynomial algorithm for the case k = 2 and hoped to use this algorithm to recognize a totally odd K4-subdivision. Our algorithm runs in O(|E(G)||V(G)|α(|E(G)|,|V(G)|)) for any fixed k. Finally, we give an O(|V(G)|2 + |E(G)|α(|E(G)|,|V(G)|log|V(G)|)) algorithm to decide whether a graph contains k disjoint paths from A to B (with |A| = |B| = k) that are not all of the same parity. This answers a conjecture of Thomassen [60]. This problem arises from the study of totally odd-K4-subdivisions in 3-connected graphs [60].