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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 31 (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.
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
R.P.: Orientations, lattice polytopes, and group arrangements II: integral and modular flow polynomials of graphs. (preprint
"... and tension polynomials of graphs ..."
Postman Problems on Mixed Graphs
, 2003
"... The mixed postman problem consists of finding a minimum cost tour of a mixed graph M = (V, E, A) traversing all its edges and arcs at least once. We prove that two wellknown linear programming relaxations of this problem are equivalent. The extra cost of a mixed postman tour T is the cost of T minu ..."
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The mixed postman problem consists of finding a minimum cost tour of a mixed graph M = (V, E, A) traversing all its edges and arcs at least once. We prove that two wellknown linear programming relaxations of this problem are equivalent. The extra cost of a mixed postman tour T is the cost of T minus the cost of the edges and arcs of M . We prove that it is to approximate the minimum extra cost of a mixed postman tour. A related
Cycles and Circuits
"... Probably the oldest and best known of all problems in graph theory centers on the bridges over the river Pregel in the city of Ko.. nigsberg (presently called Kaliningrad in Russia). The legend says that the inhabitants of Ko.. nigsberg amused themselves by trying to determine a route across each of ..."
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Probably the oldest and best known of all problems in graph theory centers on the bridges over the river Pregel in the city of Ko.. nigsberg (presently called Kaliningrad in Russia). The legend says that the inhabitants of Ko.. nigsberg amused themselves by trying to determine a route across each of the bridges between the two islands (A and B in Figure 5.1.1), both river banks (C and D of Figure 5.1.1) and back to their starting point using each bridge exactly one time. After many attempts, they all came to believe that such a route was not possible. In 1736, Leonhard Euler [15] published what is believed to be the first paper on graph theory, in which he investigated the Ko.. nigsberg bridge problem in mathematical terms.
Spanning cycles in regular matroids without M ∗ (K5) minors
"... Catlin and Jaeger proved that the cycle matroid of a 4edgeconnected graph has a spanning cycle. This result can not be generalized to regular matroids as there exist infinitely many connected cographic matroids, each of which contains a M ∗ (K5) minor and has arbitrarily large cogirth, that do not ..."
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Catlin and Jaeger proved that the cycle matroid of a 4edgeconnected graph has a spanning cycle. This result can not be generalized to regular matroids as there exist infinitely many connected cographic matroids, each of which contains a M ∗ (K5) minor and has arbitrarily large cogirth, that do not have spanning cycles. In this paper, we proved that if a connected regular matroid without a M ∗ (K5)minor has cogirth at least 4, then it has a spanning cycle. 1
Small Depth Proof Systems
"... Abstract. A proof system for a language L is a function f such that Range(f) is exactly L. In this paper, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fa ..."
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Abstract. A proof system for a language L is a function f such that Range(f) is exactly L. In this paper, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fanin circuits of constant depth (NC 0), or of O(log log n) depth but with O(1) alternations (poly log AC 0). Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local errorcorrection on a theoremproof pair. We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular language have poly log AC 0 proof systems, and from a previous result (Beyersdorff et al, MFCS 2011, where NC 0 proof systems were first introduced), this is tight. Our technique also shows that Maj has poly log AC 0 proof system. We explore the question of whether Taut has NC 0 proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that both Undirected Reachability and Directed UnReachability have NC 0 proof systems, but Directed Reachability is still open. In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of AC 0; they have SAC 0 or coSAC 0 proof systems. 1