Blerina Sinaimeri

On relaxing the constraints in pairwise compatibility graphs

A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers dmin and dmax such that each leaf lu of T corresponds to a vertex u∈V and there is an edge (u,v)∈E if and only if dmin≤dT (lu, lv)≤dmax where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where dmin=0 (LPG) and dmax=+∞ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.

Exploring pairwise compatibility graphs

A graph G=(V,E) is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge weight function w on T, and two non-negative real numbers dmi[emailxa0protected]?dmax, such that each leaf lu of T corresponds to a vertex [emailxa0protected]?V and there is an edge (u,v)@?E if and only if dmi[emailxa0protected]?dT,w(lu,lv)@?dmax where dT,w(lu,lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we analyze the class of PCGs in relation to two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves concern only dmax (LPG) or only dmin (mLPG). In particular, we show that the union of LPG and mLPG classes does not coincide with the whole class of PCGs, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, we study the closure properties of the classes PCG, mLPG and LPG, under some common graph operations. In particular, we consider the following operations: adding an isolated or universal vertex, adding a pendant vertex, adding a false or a true twin, taking the complement of a graph and taking the disjoint union of two graphs.

All Graphs with at Most Seven Vertices are Pairwise Compatibility Graphs

A graph

On types of growth for graph-different permutations

G

Pairwise Compatibility Graphs: A Survey

is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree

ON THE PAIRWISE COMPATIBILITY PROPERTY OF SOME SUPERCLASSES OF THRESHOLD GRAPHS

T

On Reverse-Free Codes and Permutations

and two non-negative real numbers

L(2,1) -labeling of oriented planar graphs

d_{min}

Rainbow graph splitting

and

On cancellative set families

d_{max}