8 research outputs found
All graphs with at most seven vertices are Pairwise Compatibility Graphs
A graph is called a pairwise compatibility graph (PCG) if there exists an
edge-weighted tree and two non-negative real numbers and
such that each leaf of corresponds to a vertex
and there is an edge if and only if where is the sum of the weights of the
edges on the unique path from to in .
In this note, we show that all the graphs with at most seven vertices are
PCGs. In particular all these graphs except for the wheel on 7 vertices
are PCGs of a particular structure of a tree: a centipede.Comment: 8 pages, 2 figure
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
Pairwise Compatibility Graphs: A Survey
International audienceA graph is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and two nonnegative real numbers and such that each leaf of is a node of and there is an edge if and only if , where is the sum of weights of the edges on the unique path from to in . In this article, we survey the state of the art concerning this class of graphs and some of its subclasses
Pairwise Compatibility Graphs of Caterpillars
A graph G=(V, E) 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 08V and there is an edge (u, v) 08E if and only if dmin 64 dT,w (lu, lv) 64 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 focus our attention on PCGs for which the witness tree is a caterpillar. We first give some properties of graphs that are PCGs of a caterpillar. We formulate this problem as an integer linear programming problem and we exploit this formulation to show that for the wheels on n vertices Wn, n=7, \u2026, 11, the witness tree cannot be a caterpillar. Related to this result, we conjecture that no wheel is PCG of a caterpillar. Finally, we state a more general result proving that any PCG admits a full binary tree as witness tree T
Pairwise Compatibility Graphs of Caterpillars
A graph G = (V, E) 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 in V and there is an edge (u, v) in E if and only if dmin <= 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 focus our attention on PCGs for which the witness tree is a caterpillar. We first give some properties of graphs that are PCGs of a caterpillar. We formulate this problem as an integer linear programming problem and we exploit this formulation to show that for the wheels on n vertices Wn, n = 7, ... , 11, the witness tree cannot be a caterpillar. Related to this result, we conjecture that no wheel is PCG of a caterpillar. Finally, we state a more general result proving that any pairwise compatibility graph admits a full binary tree as witness tree T
Pairwise Compatibility Graphs of Caterpillars
A graph G = (V, E) 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 of V and there is an edge (u, v) in E if and only if dmin <= 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 focus our attention on PCGs for which the witness tree is a caterpillar. We first give some properties of graphs that are PCGs of a caterpillar. We formulate this problem as an integer linear programming problem and we exploit this formulation to show that for the wheels on n vertices Wn, n = 7, ... , 11, the witness tree cannot be a caterpillar. Related to this result, we conjecture that no wheel is PCG of a caterpillar. Finally, we state a more general result proving that any pairwise compatibility graph admits a full binary tree as witness tree T