11 research outputs found
On relaxing the constraints in 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 paper we analyze the class
of PCG in relation with two particular subclasses resulting from the the cases
where \dmin=0 (LPG) and \dmax=+\infty (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.Comment: 12 pages, 7 figure
On 2-switches and isomorphism classes
A 2-switch is an edge addition/deletion operation that changes adjacencies in
the graph while preserving the degree of each vertex. A well known result
states that graphs with the same degree sequence may be changed into each other
via sequences of 2-switches. We show that if a 2-switch changes the isomorphism
class of a graph, then it must take place in one of four configurations. We
also present a sufficient condition for a 2-switch to change the isomorphism
class of a graph. As consequences, we give a new characterization of matrogenic
graphs and determine the largest hereditary graph family whose members are all
the unique realizations (up to isomorphism) of their respective degree
sequences.Comment: 11 pages, 6 figure
Graphs with the strong Havel-Hakimi property
The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph
to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of
zeroes produced, known as the residue, is a lower bound on the independence
number of the graph. We say that a graph has the strong Havel-Hakimi property
if in each of its induced subgraphs, deleting any vertex of maximum degree
reduces the degree sequence in the same way that the Havel-Hakimi algorithm
does. We characterize graphs having this property (which include all threshold
and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We
further show that for these graphs the residue equals the independence number,
and a natural greedy algorithm always produces a maximum independent set.Comment: 7 pages, 3 figure
Matroids arisen from matrogenic graphs
Let G be a finite simple graph and let ℐ(G) be the set of subsets X of V(G) such that the subgraph of G induced by X is threshold. If ℐ(G) is the independence system of a matroid, then G is called matrogenic [3]. In this paper, we characterize matroids arising from matrogenic graphs
Residual reliability of P-threshold graphs
We solve the problem of computing the residual reliability (the RES problem) for all classes of P-threshold graphs for which efficient structural characterizations based on decomposition to indecomposable components have been established. In particular, we give a constructive proof of existence of linear algorithms for computing residual reliability of pseudodomishold, domishold, matrogenic and matroidal graphs. On the other hand, we show that the RES problem is #P-complete on the class of biregular graphs, which implies the #P-completeness of the RES problem on the classes of indecomposable box-threshold and pseudothreshold graph
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
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
Minimal forbidden sets for degree sequence characterizations
Given a set F of graphs, a graph G is F-free if G does not contain any member of as an induced subgraph. A set F is degree-sequence-forcing (DSF) if, for each graph G in the class C of -free graphs, every realization of the degree sequence of G is also in C. A DSF set is minimal if no proper subset is also DSF. In this paper, we present new properties of minimal DSF sets, including that every graph is in a minimal DSF set and that there are only finitely many DSF sets of cardinality k. Using these properties and a computer search, we characterize the minimal DSF triples