Abstract For a graph G , two dominating sets D and $$D'$$ D ′ in non-negative integer k the set is said to -transform if there sequence $$D_0,\ldots ,D_\ell $$ 0 , … ℓ ...

Abstract Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This known as the Odd conjecture . We prove a relaxation of above namely we show admits vertex $(2t-2)$ -colouring such all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ The bound on number colours optimal ...

Dujmović et al. [J. ACM '20] proved that every planar graph is isomorphic to a subgraph of the strong product bounded treewidth and path. Analogous results were obtained for graphs Euler genus or apex-minor-free graphs. These tools have been used solve longstanding problems on queue layouts, non-repetitive colouring, p-centered adjacency labelling. This paper proves analogous structure theorems...

We provide a framework for the design and analysis of dynamic programming algorithms for H-minor-free graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2O(k·log k) · n steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedde...

We prove that for every integer t ⩾ 1 there exists a constant ct such Kt-minor-free graph G, and set S of balls in the minimum size vertices G intersecting all is at most times maximum number vertex-disjoint S. This was conjectured by Chepoi, Estellon, Vaxès 2007 special case planar graphs having same radius.

We derive precise asymptotic estimates for the number of labelled graphs not containing K3,3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K3,3-minor-free graphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs into equations for generating functions and use si...