11 research outputs found
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
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
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
The L(2,1)-labeling of unigraphs
The L(2, 1)-labeling problem consists of assigning colors from the integer set 0 ...., lambda to the nodes of a graph G in such a way that nodes at a distance of at most two get different colors, while adjacent nodes get colors which are at least two apart. The aim of this problem is to minimize lambda and it is in general NP-complete. In this paper the problem of L(2, 1)-labeling unigraphs, i.e. graphs uniquely determined by their own degree sequence up to isomorphism, is addressed and a 3/2-approximate algorithm for L(2, 1)-labeling unigraphs is designed. This algorithm runs in 0(n) time, improving the time of the algorithm based on the greedy technique, requiring 0(m) time, that may be near to Theta (n(2)) for unigraphs. (C) 2011 Elsevier B.V. All rights reserved
-product and -threshold graphs
This paper is the continuation of the research of the author and his
colleagues of the {\it canonical} decomposition of graphs. The idea of the
canonical decomposition is to define the binary operation on the set of graphs
and to represent the graph under study as a product of prime elements with
respect to this operation. We consider the graph together with the arbitrary
partition of its vertex set into subsets (-partitioned graph). On the
set of -partitioned graphs distinguished up to isomorphism we consider the
binary algebraic operation (-product of graphs), determined by the
digraph . It is proved, that every operation defines the unique
factorization as a product of prime factors. We define -threshold graphs as
graphs, which could be represented as the product of one-vertex
factors, and the threshold-width of the graph as the minimum size of
such, that is -threshold. -threshold graphs generalize the classes of
threshold graphs and difference graphs and extend their properties. We show,
that the threshold-width is defined for all graphs, and give the
characterization of graphs with fixed threshold-width. We study in detail the
graphs with threshold-widths 1 and 2
The principal Erdős–Gallai differences of a degree sequence
The Erdős–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao\u27s order on degree sequences
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
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A portfolio of theory, practice and research in a primary care setting
Counselling is a growth area in primary care. Difficulties providing a psychologically based serviced within the medical domain of the surgery have been documented, in particular the difficulties when two professionals from differing backgrounds come together to provide a service to patients. This piece of research aims to explore (both qualitatively and quantitatively) the perceptions and experiences of GPs of primary care counselling. A second aim is to compare GP's perceptions and experiences to those of primary care counsellors themselves. The objective being to help understand these difficulties (from the perspective of both professionals) as well as the factors involved in shaping a GP's perception of primary care counselling. The study focuses on the GP-counsellor relationship, GP satisfaction with the service, GPs' perceptions of counselling and the role of the counsellor in the practice. The research aims are addressed through using three different formats and combining qualitative and quantitative methods of analysis. A small group of GPs were interviewed on their experiences of a counselling service. These findings helped to shape the development of a survey looking at perceptions and experiences of primary care counselling. The survey was sent to a larger group of GPs and also to primary care counsellors. The third stage of the study involved meeting with a group of GPs to hear their views on primary care counselling services. The results obtained from the study suggest that overall GPs value their counselling service but they seem to view counselling as a passive process. There is no consensus among GPs or among counsellors as to what counselling is. The role of the counsellor within the practice does not appear to be clear to the GP and there are differences between the counsellors' perceptions of their role and the GPs' perceptions. Issues of GP power emerge from the study and many GPs believe they offer counselling to their patients. These results are developed into a model showing the factors involved in a GP's perception of primary care counselling and what factors might lead a GP to form a negative or unhelpful perception of practice based counselling. From this model interventions are suggested aimed at changing unhelpful/negative perceptions that GPs may hold and interventions designed to promote good collaborative practice. These interventions are aimed at both GPs and primary care counsellors. The study has implications for the training of both GPs and primary care counsellors as well as implications for service users. The study highlights the importance of collaborative working and the GP-counsellor relationship. Such research can help increase our understanding of inter-professional relationships