Universal Vertex Research Articles

On algorithms for ( [formula omitted],gem)-free graphs

A graph is ( P 5 ,gem)-free, when it does not contain P 5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P 4 ) as an induced subgraph. We present O ( n 2 ) time recognition algorithms for chordal gem-free graphs and for ( P 5 ,gem)-free graphs. Using a characterization of ( P 5 ,gem)-free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees [A. Brandstädt, D. Kratsch, On the structure of ( P 5 ,gem)-free graphs, Discrete Appl. Math. 145 (2005), 155–166], we give linear time algorithms for the following NP-complete problems on ( P 5 ,gem)-free graphs: Minimum Coloring; Maximum Weight Stable Set; Maximum Weight Clique; and Minimum Clique Cover.

On the structure of ( [formula omitted],gem)-free graphs

We give a complete structure description of ( P 5 ,gem)-free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P 5 is the induced path with five vertices a , b , c , d , e and four edges ab , bc , cd , de , and the gem consists of a P 4 a , b , c , d with edges ab , bc , cd plus a universal vertex e adjacent to a , b , c , d .

Circular chromatic number for iterated Mycielski graphs

For a graph G, let M( G) denote the Mycielski graph of G. The tth iterated Mycielski graph of G, M t ( G), is defined recursively by M 0( G)= G, and M t ( G)=M(M t−1 ( G)) for t⩾1. Let χ c( G) denote the circular chromatic number of G. We prove two main results: (1) If G has a universal vertex x, then χ c(M( G))= χ(M( G)) if χ c (G−x)>χ(G)− 1 2 and G is not a star, otherwise χ c ( M(G))=χ( M(G))− 1 2 ; and (2) χ c(M t ( K m ))= χ(M t ( K m )) if m⩾2 t−1 +2 t−2 and t⩾2.

GEM- AND CO-GEM-FREE GRAPHS HAVE BOUNDED CLIQUE-WIDTH

The P4 is the induced path of four vertices. The gem consists of a P4 with an additional universal vertex being completely adjacent to the P4, and the co-gem is its complement graph. Gem- and co-gem-free graphs generalize the popular class of cographs (i. e. P4-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of clique-width of graphs which extends the famous concept of treewidth. It is well-known that the cographs are exactly those graphs of clique-width at most 2. In this paper, we show that the clique-width of gem- and co-gem-free graphs is at most 16.

Local and Mean Ramsey Numbers for Some Graphs

Abstract Abstract We study local and mean Ramsey numbers for graphs with a universal vertex and some sparse graphs.

On clique-complete graphs

The clique graph, K(G), of a graph G is the intersection graph of the maximal cliques of G. For a natural number n, a graph G is n-convergent if K n ( G) is isomorphic to K 1 (the one-vertex graph). A graph G is convergent if it is n-convergent for some natural number n. A 2-convergent graph is called clique- complete. We describe the family of minimal graphs which are clique-complete but have no universal vertices. The minimality used here refers to induced subgraphs. In addition, we show that recognizing clique-complete graphs is Co-NP-complete.

On edge-colouring indifference graphs

Vizing's theorem states that the chromatic index χ′( G) of a graph G is either the maximum degree Δ( G) or Δ( G) + 1. A graph G is called overfull if ¦E(G)¦> Δ(G)∥¦V(G)¦/2∥. A sufficient condition for χ′( G) = Δ( G) + 1 is that G contains an overfull subgraph H with Δ( H) = Δ( G). Plantholt proved that this condition is necessary for graphs with a universal vertex. In this paper, we conjecture that, for indifference graphs, this is also true. As supporting evidence, we prove this conjecture for general graphs with three maximal cliques and with no universal vertex, and for indifference graphs with odd maximum degree. For the latter subclass, we prove that χ′ = Δ.

Effective field theory description of Green and vertex functions in the infrared domain.

This article discusses applications of a path-integral approach to fermionic Green and vertex functions which has a systematic expansion in terms of contours (particle ``world lines''), thereby enabling the evaluation of all-order results in the coupling constant. Our approach, provides an effective field theoretical framework for studying the low-energy regime of the underlying dynamics. In particular, the infrared behavior of the full fermion Green functions in QED can be analyzed without imposing assumptions on the asymptotic states of the charged sector, and keeping the \ensuremath{\gamma}-matrix structure of spinorial fields intact. Analytic solutions are derived for one and n fermions and the renormalization constants and associated anomalous dimensions are determined nonperturbatively; they comply with previously obtained results from exactly solvable models (e.g., Bloch-Nordsieck QED). Within an Abelian infrared setup, a universal vertex function has been calculated which embeds the soft interactions into a nontrivial contour with an infinitesimal self-intersecting loop. Adjusting the involved scales to those of the heavy quark effective theory, the result emulates the Isgur-Wise form factor and complies with the recent CLEO II data.