439 research outputs found
On dynamic threshold graphs and related classes
This paper deals with the well known classes of threshold and difference graphs, both characterized by separators, i.e. node weight functions and thresholds. We design an efficient algorithm to find the minimum separator, and we show how to maintain minimum its value when the input (threshold or difference) graph is fully dynamic, i.e. edges/nodes are inserted/removed. Moreover, exploiting the data structure used for maintaining the minimality of the separator, we study the disjoint union and the join of two threshold graphs, showing that the resulting graphs are threshold signed graphs, i.e. a superclass of both threshold and difference graphs. Finally, we consider the complement operation on all the three introduced classes of graphs.
All these operations produce in output the modified graph in terms of their separator and require time linear w.r.t. the number of different degrees. We observe that recomputing from scratch the separator would run either in linear (for threshold and difference graphs) or quadratic (for threshold signed graphs) time w.r.t. the number of nodes of the graph
Dynamically mantaining minimal integral separator for Threshold and Difference Graphs
This paper deals with the well known classes of threshold and difference graphs, both characterized by separators, i.e. node weight functions and thresholds. We show how to maintain minimum the value of the separator when the input (threshold or difference) graph is fully dynamic, i.e. edges/nodes are inserted/removed. Moreover, exploiting the data structure used for maintaining the minimality of the separator, we handle the operations of disjoint union and join of two threshold graphs. © Springer International Publishing Switzerland 2016
On conjectures and problems of Ruzsa concerning difference graphs of S-units
Given a finite nonempty set of primes S, we build a graph with
vertex set by connecting x and y if the prime divisors of both the
numerator and denominator of x-y are from S. In this paper we resolve two
conjectures posed by Ruzsa concerning the possible sizes of induced
nondegenerate cycles of , and also a problem of Ruzsa concerning
the existence of subgraphs of which are not induced subgraphs.Comment: 15 page
Representation of finite graphs as difference graphs of S-units, I
In part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v_1,v_2 are connected by an edge if and only if the difference of the attached values is an S-unit. In part I we gave several results concerning the representability of graphs in the above sense.In the present paper we extend the results from paper I to the algebraic number field case and make some of them effective. Besides we prove some new theorems: we prove that G is infinitely representable with S if and only if it has a degenerate representation with S, and we also deal with the representability with S of the union of two graphs of which at least one is finitely representable with S.p, li { white-space: pre-wrap; }</style
-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
Difference Ramsey Numbers and Issai Numbers
We present a recursive algorithm for finding good lower bounds for the
classical Ramsey numbers. Using notions from this algorithm we then give some
results for generalized Schur numbers, which we call Issai numbers.Comment: 10 page
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