Size Of Subgraph Research Articles

Cluster fault‐tolerant routing in star graphs

Fault-tolerant routing is a key issue in computer/ communication networks. We say a network (graph) can tolerate I faulty nodes for a routing problem If after removing at most / arbitrary faulty nodes from the graph the routing paths exist for the routing problem. However, the bound / is usually a worst-case measure and It is of great interest to find the routing paths when more than / faulty nodes exist. Cluster fault-tolerant (CFT) routing was proposed as an approach for this purpose. In the CFT routing, we reduce the number of faults that a routing problem has to deal with using subgraphs to cover the faulty nodes. In particular, we consider the number and the size (diameter) of faulty subgraphs rather than the number of faulty nodes that a graph can tolerate. In this paper, we show that a subgraph of diameter 2 can be viewed as a single fault for the following routing problems in the star graph: Given a source node s and t target nodes t 1 , ..., t k , find k node-disjoint paths from s to t 1 (1 ≤ i ≤ k), and given k node pairs (s 1 , t 1 ), (s k , t k ), find k node-disjoint paths s f → t i (1 ≤ i ≤ k). Since a subgraph of diameter 2 of the n-dimensional star graph G n may have n nodes, the above result implies that the number of faulty nodes that G n can tolerate is n times larger than the worst-case measure if the faulty nodes can be covered by certain subgraphs. We also give algorithms which find the paths for the two routing problems.

Cluster fault-tolerant routing in star graphs

Fault-tolerant routing is a key issue in computer/ communication networks. We say a network (graph) can tolerate I faulty nodes for a routing problem If after removing at most / arbitrary faulty nodes from the graph the routing paths exist for the routing problem. However, the bound / is usually a worst-case measure and It is of great interest to find the routing paths when more than / faulty nodes exist. Cluster fault-tolerant (CFT) routing was proposed as an approach for this purpose. In the CFT routing, we reduce the number of faults that a routing problem has to deal with using subgraphs to cover the faulty nodes. In particular, we consider the number and the size (diameter) of faulty subgraphs rather than the number of faulty nodes that a graph can tolerate. In this paper, we show that a subgraph of diameter 2 can be viewed as a single fault for the following routing problems in the star graph: Given a source node s and t target nodes t 1 , ..., t k , find k node-disjoint paths from s to t 1 (1 ≤ i ≤ k), and given k node pairs (s 1 , t 1 ), (s k , t k ), find k node-disjoint paths s f → t i (1 ≤ i ≤ k). Since a subgraph of diameter 2 of the n-dimensional star graph G n may have n nodes, the above result implies that the number of faulty nodes that G n can tolerate is n times larger than the worst-case measure if the faulty nodes can be covered by certain subgraphs. We also give algorithms which find the paths for the two routing problems.

Exact Bounds for Judicious Partitions of Graphs

has a bipartite subgraph of size at least . We show that every graph of size has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most edges. This is exact for complete graphs of odd order, which we show are the only extremal graphs without isolated vertices. We also give results for partitions into more than two classes.

Tight Bounds for the Maximum Acyclic Subgraph Problem

Given a directed graphG=(V,A), themaximum acyclic subgraph problemis to find a maximum cardinality subsetA′ of the arcs such thatG′=(V,A′) is acyclic. In this paper, we present polynomial-time and RNC algorithms which, when given any graphGwithout two-cycles, find an acyclic subgraph of size at least(1/2+Ω(1/Δ(G)))|A|, where Δ(G) is the maximum degree ofG. This bound is existentially tight, since there exists a class of graphs without two-cycles for which the largest acyclic subgraph has size at most(1/2+O(1/Δ(G)))|A|. For the common case of low-degree graphs, our algorithms provide an even better improvement over the known algorithms that find an acyclic subgraph of size 12|A|. For example, for graphs without two-cycles, our algorithms find an acyclic subgraph of size at least23|A|when Δ(G)=2 or 3, and1930|A|when Δ(G)=4 or 5. For Δ(G)=2, this bound is optimal in the worst case. For 3-regular graphs, we can achieve1318|A|, which is slightly better than frac23;|A|. As a consequence of this work, we find that all graphs without two-cycles contain large acyclic subgraphs, a fact which was not previously known. The results can also be extended to find large acyclic subgraphs of graphs with two-cycles.

EXPANDING GRAPH RELABELING SYSTEMS HAVE THE POWER OF RECURSIVE ENUMERABILITY

Graph relabeling systems (GRS's) have been introduced as a suitable tool for coding and proving sequential or distributed algorithms on graphs or networks. These systems do not change the underlying structure of the graph on which they work, but only the labeling of its components (edges or vertices). Each relabeling step is fully determined by the knowledge of a fixed size subgraph, the relabeled occurrence. We introduce an extension of that model, the so-called expanding graph relabeling systems (e-GRS's), which allows the generation of sets of graphs by means of component relabeling. We study the generating power of these systems and prove that they enable us to generate any recursively enumerable set of graphs. We first show how the “from left to right” natural orientation of a string-graph, that is a graph representation of a string, can be translated by means of vertex labels in such a way that any local transformation of the string can be simulated by a local relabeling of the string-graph vertices. Using this translation, we show that any phrase-structure string grammar can be simulated by an e-GRS. Finally, we provide a way of encoding graphs as strings and an e-GRS, called the decoder, which can convert any string representation of the encoding of a graph into the graph itself.

Bisection of trees and sequences

A graph G is called bisectable if it is an edge-disjoint union of two isomorphic subgraphs. We show that any tree T with e edges contains a bisectable subgraph with at least e−O( e/loglog e) edges. We also show that every forest of size e, each component of which is a star, contains a bisectable subgraph of size at least e−O(log 2 e).

On Subgraph Sizes in Random Graphs

For a graph G with m edges let its Range of Subgraph Sizes (RSS)ρ(G) = {t : G contains a vertex-induced subgraph with t edges}.G has a full RSS if ρ(G) = {0, 1, …, m}. We establish the threshold for a random graph to have a full RSS and give tight bounds on the likely RSS of a dense random graph.

The size of chordal, interval and threshold subgraphs

Given a graphG withn vertices andm edges, how many edges must be in the largest chordal subgraph ofG? Form=n2/4+1, the answer is 3n/2−1. Form=n2/3, it is 2n−3. Form=n2/3+1, it is at least 7n/3−6 and at most 8n/3−4. Similar questions are studied, with less complete results, for threshold graphs, interval graphs, and the stars on edges, triangles, andK4's.

Ki‐covers. II. Ki‐perfect graphs

AbstractAKi is a complete subgraph of size i. A Ki‐cover of a graph G(V, E) is a set C of Ki−1s of G such that every Ki in G contains at least one Ki−1 in C. ci(G) is the cardinality of a smallest Ki‐cover of G. A Ki‐packing of G is a set of Kis such that no two Kis have i − 1 nodes in common. pi(G) is the cardinality of a largest Ki‐packing of G. Let Fi(G) denote the set of Kis in G and define ci(F) and pi(F) analogously for F ⊆ Fi(G). G is Ki‐perfect if ∀F ⊆ Fi(G), ci(F) = pi(F). The K2‐perfect graphs are precisely the bipartite graphs. We present a characterization of Ki‐perfect graphs that is similar to the Strong Perfect Graph Conjecture, and explore the relationships between Ki‐perfect graphs and normal hypergraphs. Furthermore, if iA denotes the 0 − 1 matrix of G where the rows are the elements of Fi−1(G) that belong to at least one Ki and the columns are the elements of Fi(G), then we show that iA is perfect iff G is a Ki‐perfect graph. We also characterize the Ki‐perfect graphs for which iAis balanced.

Packings by Complete Bipartite Graphs

Given any set $\mathcal{B}$ of complete bipartite graphs, we ask whether a graph H admits a $\mathcal{B}$-factor, i.e., a spanning subgraph, each of whose components is a member of $\mathcal{B}$. More generally, we seek in H a maximum $\mathcal{B}$-packing, i.e., a $\mathcal{B}$-factor of a maximum size subgraph of H. We first treat the interesting special case when $\mathcal{B}$ is a set of stars. The results are generalized to arbitrary $\mathcal{B}$ in the last section. We prove for most of these problems that they are $\mathcal{N}\mathcal{P}$-hard; we also show that the remaining problems admit polynomial algorithms based on augmenting configurations. The simplicity of these algorithms, as well as the implied min-max theorems, resemble the theory of matchings in bipartite, rather than general, graphs.

Ki-covers I: Complexity and polytopes

A K i in a graph is a complete subgraph of size i. A K i -cover of a graph G( V, E is a set C of K i − 1 's of G such that every K i in G contains at least one K i − 1 in C. Thus a K 2-cover is a vertex cover. The problem of determining whether a graph has a K i -cover ( i ⩾ 2) of cardinality ⩽ k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the K i -cover polytope. Further results of this type are also produced for the K 3-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the K i -cover polytope.

Interpolation theorem for the number of pendant vertices of connected spanning subgraphs of equal size

At the 4th International Graph Theory Conference 1980, G. Chartrand posed the following problem: If a (connected) graph G contains spanning trees with m and n pendant vertices, respectively, with m < n, does G contain a spanning tree with k pendant vertices for every integer k, where m < k < n? Recently, S. Schuster showed that the answer is yes. Several variations of this interpolation theorem will be given including the following generalization: If a connected graph G contains connected spanning subgraphs of size r with m and n pendant vertices, respectively, with m < n, then G contains a connected spanning subgraph of size r with k pendant vertices for every integer k, where m < k < n.

On the chromatic number of subgraphs of a given graph

It is shown that for arbitrary positive integers m,n there exists a ϕ ( m , n ) \phi (m,n) such that if χ ( G ) ⩾ ϕ ( m , n ) \chi (G) \geqslant \phi (m,n) , then G contains either a complete subgraph of size m or a subgraph H with χ ( H ) = n \chi (H) = n containing no C 3 {C_3} . This gives an answer to a problem of Erdös and Hajnal.