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11
Protection Cycles in Mesh WDM Networks
 IEEE Journal on Selected Areas in Communications
, 2000
"... A fault recovery system that is fast and reliable is essential to today's networks, as it can be used to minimize the impact of the fault on the operation of the network and the services it provides. This paper proposes a methodology for performing automatic protection switching (APS) in optical net ..."
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Cited by 39 (0 self)
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A fault recovery system that is fast and reliable is essential to today's networks, as it can be used to minimize the impact of the fault on the operation of the network and the services it provides. This paper proposes a methodology for performing automatic protection switching (APS) in optical networks with arbitrary mesh topologies in order to protect the network from fiber link failures. All fiber links interconnecting the optical switches are assumed to be bidirectional. In the scenario considered, the layout of the protection fibers and the setup of the protection switches is implemented in nonreal time, during the setup of the network. When a fiber link fails, the connections that use that link are automatically restored and their signals are routed to their original destination using the protection fibers and protection switches. The protection process proposed is fast, distributed, and autonomous. It restores the network in real time, without relying on a central manager or a centralized database. It is also independent of the topology and the connection state of the network at the time of the failure.
Supereulerian graphs: A survey
 J. Graph Theory
, 1992
"... A graph is supereulerian if it has a spanning eulerian subgraph. There is a reduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a grap ..."
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Cited by 30 (4 self)
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A graph is supereulerian if it has a spanning eulerian subgraph. There is a reduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a graph. We outline the research on supereulerian graphs, the reduction method and its applications. 1. Notation We follow the notation of Bondy and Murty [22], with these exceptions: a graph has no loops, but multiple edges are allowed; the trivial graph K1 is regarded as having infinite edgeconnectivity; and the symbol E will normally refer to a subset of the edge set E(G) of a graph G, not to E(G) itself. The graph of order 2 with 2 edges is called a 2cycle and denoted C2. Let H be a subgraph of G. The contraction G/H is the graph obtained from G by contracting all edges of H and deleting any resulting loops. For a graph G, denote O(G) = {odddegree vertices of G}. A graph with O(G) = ∅ is called an even graph. A graph is eulerian if it is connected and even. We call a graph G supereulerian if G has a spanning eulerian subgraph. Regard K1 as supereulerian. Denote SL = {supereulerian graphs}. 1 Let G be a graph. The line graph of G (called an edge graph in [22]) is denoted L(G), it has vertex set E(G), where e, e ′ ∈ E(G) are adjacent vertices in L(G) whenever e and e ′ are adjacent edges in G. Let S be a family of graphs, let G be a graph, and let k ≥ 0 be an integer. If there is a graph G0 ∈ S such that G can be obtained from G0 by removing at most k edges, then G is said to be at most k edges short of being in S. For a graph G, we write F (G) = k if k is the least nonnegative integer such that G is at most k edges short of having 2 edgedisjoint spanning trees. 2.
Graphs with the circuit cover property
, 1994
"... A circuit cover of an edgeweighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edgecut is even, and no edge has more than half the t ..."
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Cited by 23 (2 self)
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A circuit cover of an edgeweighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edgecut is even, and no edge has more than half the total weight of any edgecut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p. We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2edgeconnected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.
NowhereZero 3Flows in Squares of Graphs
, 2003
"... It was conjectured by Tutte that every 4edgeconnected graph admits a nowherezero 3flow. In this paper, we give a complete characterization of graphs whose squares admit nowherezero 3flows and thus confirm Tutte's 3flow conjecture for the family of squares of graphs. ..."
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Cited by 4 (3 self)
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It was conjectured by Tutte that every 4edgeconnected graph admits a nowherezero 3flow. In this paper, we give a complete characterization of graphs whose squares admit nowherezero 3flows and thus confirm Tutte's 3flow conjecture for the family of squares of graphs.
MOD (2p + 1)ORIENTATIONS AND K1,2p+1DECOMPOSITIONS
, 2007
"... In this paper, we establish an equivalence between the contractible graphs with respect to the mod (2p + 1)orientability and the graphs with K1,2p+1decompositions. This is applied to disprove a conjecture proposed by Barat and Thomassen that every 4edgeconnected simple planar graph G with E(G) ..."
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Cited by 2 (0 self)
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In this paper, we establish an equivalence between the contractible graphs with respect to the mod (2p + 1)orientability and the graphs with K1,2p+1decompositions. This is applied to disprove a conjecture proposed by Barat and Thomassen that every 4edgeconnected simple planar graph G with E(G) ≡ 0 (mod 3) has a claw decomposition.
Extending a Partial NowhereZero 4Flow
, 1999
"... ... every graph with 2 edgedisjoint spanning trees admits a nowherezero 4flow. In [J Combin Theory Ser B, 56 (1992), 165–182], Jaeger et al. extended this result by showing that, if A is an abelian group with A  =4, then every graph with 2 edgedisjoint spanning trees is Aconnected. As graphs ..."
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Cited by 2 (2 self)
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... every graph with 2 edgedisjoint spanning trees admits a nowherezero 4flow. In [J Combin Theory Ser B, 56 (1992), 165–182], Jaeger et al. extended this result by showing that, if A is an abelian group with A  =4, then every graph with 2 edgedisjoint spanning trees is Aconnected. As graphs with 2 edgedisjoint spanning trees are all collapsible, we in this note improve the latter result by showing that, if A is an abelian group with A  =4, then every collapsible graph is Aconnected. This allows us to prove the following generalization of Jaeger’s theorem: Let G be a graph with 2 edgedisjoint spanning trees and let M be an edge cut of G with M ≤4. Then either any partial nowherezero 4flow on M can be extended to a nowherezero 4flow of the whole graph G, or G can be contracted to one of three configurations, including the wheel of 5 vertices, in which cases certain partial nowherezero 4flows on M cannot be extended. Our results also improve a theorem of Catlin in [J Graph Theory, 13 (1989), 465–483].
Supereulerian Graphs and the Petersen Graph
, 1990
"... Any 3edgeconnected graph with at most 10 edge cuts of size 3 either has a spanning closed trail or it is contractible to the Petersen graph. 1996 Academic Press, Inc. Introduction and Notation We shall follow the notation of Bondy and Murty [1], but with minor variations. An arc in a graph G is a ..."
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Cited by 1 (1 self)
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Any 3edgeconnected graph with at most 10 edge cuts of size 3 either has a spanning closed trail or it is contractible to the Petersen graph. 1996 Academic Press, Inc. Introduction and Notation We shall follow the notation of Bondy and Murty [1], but with minor variations. An arc in a graph G is a path in G whose internal vertices have degree 2 in G. Denote O(G)=[odddegree vertices of G].
Graph family operations
, 2001
"... ... sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero kflows for a fixed integer k >= 3. Unfortunately, a subtle error caused several theorems previously published in C ..."
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Cited by 1 (1 self)
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... sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero kflows for a fixed integer k >= 3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.
Double cycle covers and the Petersen graph, III
"... Any graph with no cut edge and with at most 13 edge cuts of size 3 either has a cycle double cover formed by three subgraphs with eulerian components, or it is contractible to the Petersen graph. We correct an error in an earlier proof of this result. We allow loops but not multiple edges. Define O( ..."
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Any graph with no cut edge and with at most 13 edge cuts of size 3 either has a cycle double cover formed by three subgraphs with eulerian components, or it is contractible to the Petersen graph. We correct an error in an earlier proof of this result. We allow loops but not multiple edges. Define O(G) to be the set of odddegree vertices of G. A graph is even if O(G) = ∅. Let S3 denote the family of graphs for which there is a partition E(G) = E1 ∪E2 ∪E3 such that O(G[Ei]) = O(G) (1 ≤ i ≤ 3). For 3regular graphs, S3 is the family of graphs having a 1factorization. The Petersen graph, denoted P, is the smallest 2edgeconnected graph not in S3. Since we are simply correcting an erroneous proof of a lemma of a prior paper (Lemma 13 of [1]), we shall confine this note to a proof of that lemma, plus background for the proof. For more details, see [1]. Jaeger [2] proved the following result for graphs with no edge cut of size 1 or 3. Theorem 1 Let G be a graph with no cut edge. If G has at most 13 edge cuts of size 3, then exactly one of these holds: (a) G ∈ S3; (b) G is contractible to P. For any graph G with nonparallel edges xy and yz incident with a vertex y, where d(y) ≥ 3, the graph G0 (say) obtained from G−{xy, yz} by adding a new edge xz is said to be obtained from G by lifting {xy, yz}; and that pair of edges is said to be lifted to form G0. Theorem 2 (Mader [3]) Suppose that y ∈ V (G) is not a cutvertex of G. If d(y) ≥ 4, then some pair of edges incident with y can be 1 lifted, so that in the resulting graph G0, any pair of distinct vertices v, w ∈ V (G) − y satisfy κ ′ G0 (v, w) = κ ′ G(v, w).
Collapsible Graphs and Reduction of Line Graphs 1 Collapsible Graphs and Reduction of Line Graphs
"... A graph G is collapsible if for every even subset X ⊆ V (G), G has a spanning connected subgraph whose set of odddegree vertices is X. A graph obtained by contracting all the nontrivial collapsible subgraphs of G is called the reduction of G. In this paper, we characterize graphs of diameter two in ..."
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A graph G is collapsible if for every even subset X ⊆ V (G), G has a spanning connected subgraph whose set of odddegree vertices is X. A graph obtained by contracting all the nontrivial collapsible subgraphs of G is called the reduction of G. In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [Journal of Graph Theory, Vol. 14, No. 1, 7787 (1990)], and in [Journal of Graph Theory, Vol. 14 (1990) 347364].