A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The list of minimal forbidden induced subgraphs for the c...

In this paper, we prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M(K4).

The G-width of a class of graphs G is defined as follows. A graph G has G-width k if there are k independent sets N1, . . . ,Nk in G such that G can be embedded into a graph H ∈ G such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-para...

Given a class of graphs F , a forbidden subgraph characterization (FSC) is a set of graphs H such that a graph G belongs to F if and only if no graph of H is isomorphic to an induced subgraph of G. FSCs play a key role in graph theory, and are at the center of many important results obtained in that field. In this paper, we present novel methods that automate the generation of conjectures on FS...

For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive...

Given a graph G = (V; E), the metric polytope S(G) is deened by the inequalities x(F) ? x(C n F) jFj ? 1 for F C; jFj odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the max-cut problem. The graph G is called 1 d-integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d-integral graphs is closed ...

Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. For a graph class G, the price of connectivity for feedback vertex set (poc-fvs) for G is defined as the maximum ratio cfvs(G)/fvs(G) over all connected graphs G in G. It is known that the poc-fvs for general graphs is unbounded. We study the p...

Group-IV nanocrystals have emerged as a promising group of materials that extends the realm of application of bulk diamond, silicon, germanium and related materials beyond their traditional boundaries. Over the last two decades of research, their potential for application in areas such as optoelectronic applications and memory devices has been progressively unraveled. Nevertheless, new challeng...

We characterize all pairs of connected graphs {X,Y } such that each 3-connected {X,Y }-free graph is pancyclic. In particular, we show that if each of the graphs in such a pair {X,Y } has at least four vertices, then one of them is the claw K1,3, while the other is a subgraph of one of six specified graphs.

A circular-arc model (C,A) is a circle C together with a collection A of arcs of C. If A satisfies the Helly Property then (C,A) is a Helly circular-arc model. A (Helly) circular-arc graph is the intersection graph of a (Helly) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention, in the literature. Linear time recognition algorithm have...