In [6], we introduced the notion of minor-embedding in adiabatic quantum optimization. A minor-embedding of a graph G in a quantum hardware graph U is a subgraph of U such that G can be obtained from it by contracting edges. In this paper, we describe the intertwined adiabatic quantum architecture design problem, which is to construct a hardware graph U that satisfies all known physical constra...

Let G be a cubic graph and the graph 2G is obtained by replacing each edge of G with a pair of parallel edges. A proper 6-edgecoloring of 2G is called a Fulkerson coloring of G. It was conjectured by Fulkerson that every bridgeless cubic graph has a Fulkerson coloring. In this paper we show that for a Petersen-minor free Graph G, G is uniquely Fulkerson colorable if and only if G constructed fr...

We examined the effects of interleukin-3 (IL-3) and c-kit ligand (KL) on the survival of differentiated hematopoietic progenitor cells (HPC), the burst-forming unit-erythroid (BFU-E); colony-forming unit-granulocyte, erythroid, monocyte, megakaryocyte (CFU-GEMM); and CFU-granulocyte-monocyte (CFU-GM) and more primitive hematopoietic cells that give rise to these progenitor cells (pre-colony-for...

We improve the running time of the general algorithmic technique known as Baker’s approach (1994) on H-minor-free graphs from O(n) to O(f(|H |)n) showing that it is fixed-parameter tractable w.r.t. the parameter |H |. The numerous applications include e.g. a 2-approximation for coloring and PTASes for various problems such as dominating set and max-cut, where we obtain similar improvements. On ...

We prove two results: 1. A graph G on at least seven vertices with a vertex v such that G − v is planar and t triangles satisfies |E(G)| ≤ 3|V (G)| − 9 + t/3. 2. For p = 2, 3, . . . , 9, a triangle-free graph G on at least 2p − 5 vertices with no Kp-minor satisfies |E(G)| ≤ (p− 2)|V (G)| − (p− 2).

The minor crossing number of a graph G is the minimum crossing number of a graph that contains G as a minor. It is proved that for every graph H there is a constant c, such that every graph G with no H-minor has minor crossing number at most c|V (G)|.

A graph H is t-apex if H − X is planar for some set X ⊂ V (H) of size t. For any integer t ≥ 0 and a fixed t-apex graph H, we give a polynomial-time algorithm to decide whether a (t + 3)-connected Hminor-free graph is colorable from a given assignment of lists of size t+4. The connectivity requirement is the best possible in the sense that for every t ≥ 1, there exists a t-apex graph H such tha...

A block, considered as a set of elements together with its adjacency matrix M, is called a B-block if M is the adjacency matrix of a complete bipartite graph Kk,, k, . A balanced bipartite design with parameters b, u, r, k, h, kl, k2 (briefly BBD(u, k, A; kl)) is an arrangement of u elements into b B-blocks such that every B-block contains k = kl + ki elements, every element occurs in exactly r...

The Hadwiger number of a graph G is the largest integer h such that G has the complete graph Kh as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h suc...