We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer k-planar graphs, where each edge is crossed by at most k other edges; and, outer k-quasi-planar graphs where no k edges can mutually cross. We show that the outer k-planar graphs are (b √ 4k + 1...

A general method for the computation of various parameters measuring the vulnerability of a graph is introduced. Four measures of vulnerability are considered, i.e., the toughness, scattering number, vertex integrity and the size of a minimum balanced separator. We show how to compute these parameters by polynomial time algorithms for various classes of intersection graphs like permutation grap...

Usual graph classes, such as complete graphs, paths, cycles and stars, frequently appear as extremal graphs in graph theory problems. Here we want to turn the reader’s attention to two novel, simply defined, graph classes that appear as extremal graphs in several graph theory problems. We call them bags and bugs. As examples of problems where bags and bugs appear, we show that balanced bugs max...

We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a nonsplit link for every signed embedding. With one ...

We present polynomial time algorithms for induced biclique optimization problems in the following families of graphs: polygon-circle graphs, 4-hole-free graphs, complements of intervalfilament graphs and complements of subtree-filament graphs. Such problems are to find maximum: induced bicliques, induced balanced bicliques and induced edge bicliques. These problems have applications for bicliqu...

............................................................................................................................................ 2 ACKNOWLEDGEMENTS ....................................................................................................................... 4 CONTENTS .............................................................................................................

A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it equivalent to an unsigned under operation of sign switching; otherwise called sign-unbalanced. nut a one dimensional kernel with corresponding eigenvector that full. In this paper we generalise notion graphs graphs. Orders for which regular exist were determined recently degrees up $11$. By extendi...

An equitable k-coloring of a graph G is proper such that the sizes any two color classes differ by at most one. (q,r)-tree-coloring an q-coloring subgraph induced each class forest maximum degree r. Let strong vertex r-arboricity G, denoted var≡(G), be minimum p has for every q≥p. The values va1≡(Kn,n) were investigated Tao and Lin Wu, Zhang, Li where exact found in some special cases. In this ...

In this paper we propose and study a new structural invariant for graphs, called distance-unbalancedness, as measure of how much graph is (un)balanced in terms distances. Explicit formulas are presented several classes well-known graphs. Distance-unbalancedness trees also studied. A few conjectures stated some open problems proposed.