The MaximumWeight Independent Set (MWIS) problem is a well-known NPhard problem. A popular way to study MWIS is to detect graph classes for which MWIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. For graphs G1, G2, G1 + G2 denotes the disjoint uni...

The toughness of a (non-complete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields |A|/t components. Determining toughness is an NP-hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least 3 vertices with toughness at least t is hamiltonian, is still open for ...

The chain graph sandwich problem asks: Given a vertex set V , a mandatory edge set E1, and a larger edge set E2, is there a graph G= (V ,E) such that E1 ⊆ E ⊆ E2 with G being a chain graph (i.e., a 2K2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E1 is a connected graph, which has a linear-time solutio...

We define the A4-structure of a graph G to be the 4-uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K2, C4, or P4. We show that perfection of a graph is determined by its A4-structure. We relate the A4structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Mathematics 220 (2000) 201-238]; for example, a graph is indecomposabl...

The Ramsey number r(G,H) is determined for all disconnected (isolatefree) graphs H of order six and all graphs G of order at most five, except the three cases (G,H) ∈ {(K5 − 2K2, 2K3), (K5 − e, 2K3), (K5, 2K3)} where bounds with difference 1 are established. Moreover, general results are obtained for some small disconnected graphs H and any graph G.

For a connected graph G = (V,E), a subset U ⊆ V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NPcomplete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing if a graph allows a vertex-surjectiv...

For a connected graph G = (V,E), a subset U ⊆ V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NPcomplete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing if a graph allows a vertex-surjectiv...

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating diffi...

Let Km −H be the graph obtained from Km by removing the edges set E(H) of H where H is a subgraph of Km. In this paper, we characterize the potentially K5 − P3, K5 − A3, K5 −K3 and K5 − K1,3-graphic sequences where A3 is P2 ∪ K2. Moreover, we also characterize the potentially K5 − 2K2-graphic sequences where pK2 is the matching consisted of p edges.

The modulation transfer function, noise transfer function and detection quantum efficiency of both 1k2 and 2k2 cameras attached to the same 300 kV FEGTEM have been measured. The results show that both the MTF and the DQE of the 2k2 camera are lower than for the 1k2 camera at frequencies above a third of the Nyquist limit. This lower per-pixel performance of the 2k2 camera is primarily a result ...