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31.
The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the
signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also
prove that every non-empty set of integers is the signed degree set of some connected signed graph. 相似文献
32.
E. Alper Yildirim Xiaofei Fan-Orzechowski 《Computational Optimization and Applications》2006,33(2-3):229-247
We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions
of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph
to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable
set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite
formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex
that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure
is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly
rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start
strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently
extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality.
This work was supported in part by NSF through CAREER Grant DMI-0237415. Part of this work was performed while the first author
was at the Department of Applied Mathematics and Statisticsat Stony Brook University, Stony Brook, NY, USA. 相似文献
33.
Vasiliy A. Ustimenko 《Acta Appl Math》2002,74(2):117-153
A combinatorial method of encryption with a similarity to the classical scheme of linear coding has been suggested by the author. The general idea is to treat vertices of a graph as messages and arcs of a certain length as encryption tools. We will study the quality of such an encryption in the case of graphs of high girth by comparing the probability to guess the message, (vertex) at random with the probability of breaking the key, i.e. guessing the encoding arc. In fact, the quality is good for graphs which are close to the Erdös bound, defined by the Even Cycle Theorem.In the case of parallelotopic graphs, there is a uniform way to match arcs with strings in a certain alphabet. Among parallelotopic graphs we distinguish linguistic graphs of affine type whose vertices (messages) and arcs (encoding tools) both could be naturally identified with vectors over the GF(q), and neighbors of the vertex defined by a system of linear equations. We will discuss families of linguistic and parallelotopic graphs of increasing girth as the source for assymmetric cryptographic functions and related open key algorithms.Several constructions of families of linguistic graphs of high girth with good quality, complexity and expansion coefficients will be considered. Some of those constructions have been obtained via group-theoretical and geometrical techniques. 相似文献
34.
Given a rectangular array whose entries represent the pixels of a digitalized image, we consider the problem of reconstructing an image from the number of occurrences of each color in every column and in every row. The complexity of this problem is still open when there are just three colors in the image. We study some special cases where the number of occurrences of each color is limited to small values. Formulations in terms of edge coloring in graphs and as timetabling problems are used; complexity results are derived from the model. 相似文献
35.
Thomas Andreae 《Journal of Graph Theory》2002,39(4):222-229
For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ?0 disjoint copies of A is referred to as ?0A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all n??, but ?0A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G1, G2,… such that Gi is not a minor of Gj for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ? ?, then ?0A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016 相似文献
36.
记Ore2=min{d(y) d(x)|x,y∈V(G),d(x,y)=2},本得到:若n阶图G的Ore2≥n 1,则G是[5;n]泛连通图。此是比Faudree等人的定理进一步的结果。 相似文献
37.
38.
Chepoi showed that every breadth first search of a bridged graph produces a cop-win ordering of the graph. We note here that Chepoi's proof gives a simple proof of the theorem that G is bridged if and only if G is cop-win and has no induced cycle of length four or five, and that this characterization together with Chepoi's proof reduces the time complexity of bridged graph recognition. Specifically, we show that bridged graph recognition is equivalent to (C4,C5)-free graph recognition, and reduce the best known time complexity from O(n4) to O(n3.376). 相似文献
39.
A (p, q)-sigraph S is an ordered pair (G, s) where G = (V, E) is a (p, q)-graph and s is a function which assigns to each edge of G a positive or a negative sign. Let the sets E
+ and E
– consist of m positive and n negative edges of G, respectively, where m + n = q. Given positive integers k and d, S is said to be (k, d)-graceful if the vertices of G can be labeled with distinct integers from the set {0, 1, ..., k + (q – 1)d such that when each edge uv of G is assigned the product of its sign and the absolute difference of the integers assigned to u and v the edges in E
+ and E
– are labeled k, k + d, k + 2d, ..., k + (m – 1)d and –k, – (k + d), – (k + 2d), ..., – (k + (n – 1)d), respectively.In this paper, we report results of our preliminary investigation on the above new notion, which indeed generalises the well-known concept of (k, d)-graceful graphs due to B. D. Acharya and S. M. Hegde. 相似文献
40.
Let G = (V,E) be a graph or digraph and r : V → Z+. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν1,…,νr(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003 相似文献