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28
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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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.
Graph learning with a nearest neighbor approach
 In Proceedings of the Conference on Computational Learning Theory
, 1996
"... In this paper, we study how to traverse all edges of an unknown graph G =(V; E) that is bidirected and strongly connected. This problem can be solved with a simple algorithm that traverses all edges at most twice, and no algorithm can do better in the worst case. Artificial Intelligence researchers ..."
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Cited by 18 (6 self)
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In this paper, we study how to traverse all edges of an unknown graph G =(V; E) that is bidirected and strongly connected. This problem can be solved with a simple algorithm that traverses all edges at most twice, and no algorithm can do better in the worst case. Artificial Intelligence researchers, however, often use the following online nearest neighbor algorithm: “repeatedly take a shortest path to the closest unexplored edge and traverse it. ” We prove bounds on the worstcase complexity of this algorithm. We show, for example, that its worstcase complexity is close to optimal for some classes of graphs, such as graphs with linear or star topology and dense graphs with edge lengths one. In general, however, its complexity can grow faster than linear in the sum of all edge lengths, although not faster than log(V) times the sum of all edge lengths. 1
Speeding Up Evolutionary Algorithms through Restricted Mutation Operators
 In Proc. of PPSN ’06
, 2006
"... Abstract. We investigate the effect of restricting the mutation operator in evolutionary algorithms with respect to the runtime behavior. For the Eulerian cycle problem; we present runtime bounds on evolutionary algorithms with a restricted operator that are much smaller than the best upper bounds f ..."
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Abstract. We investigate the effect of restricting the mutation operator in evolutionary algorithms with respect to the runtime behavior. For the Eulerian cycle problem; we present runtime bounds on evolutionary algorithms with a restricted operator that are much smaller than the best upper bounds for the general case. It turns out that a plateau that both algorithms have to cope with is left faster by the new algorithm. In addition, we present a lower bound for the unrestricted algorithm which shows that the restricted operator speeds up computation by at least a linear factor. 1
Easy and hard testbeds for realtime search algorithms
 In Proceedings of the National Conference on Artificial Intelligence
, 1996
"... Although researchers have studied which factors influence the behavior of traditional search algorithms, currently not much is known about how domain properties influence the performance of realtime search algorithms. In this paper we demonstrate, both theoretically and experimentally, that Euleria ..."
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Cited by 14 (8 self)
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Although researchers have studied which factors influence the behavior of traditional search algorithms, currently not much is known about how domain properties influence the performance of realtime search algorithms. In this paper we demonstrate, both theoretically and experimentally, that Eulerian state spaces(a supersetof undirected state spaces)are very easy for some existing realtime search algorithms to solve: even realtime search algorithms that can be intractable, in general, are efficient for Eulerian state spaces. Because traditional realtime search testbeds (such as the eight puzzle and gridworlds) are Eulerian, they cannot be used to distinguish between efficient and inefficient realtime search algorithms. It follows that one has to use nonEulerian domains to demonstrate the general superiority of a given algorithm. To this end, we present two classes of hardtosearch state spaces and demonstrate the performance of various realtime search algorithms on them.
Parameterized Complexity of Vertex Cover Variants
, 2006
"... Important variants of the Vertex Cover problem (among others, Connected Vertex Cover, Capacitated Vertex Cover, and Maximum ..."
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Important variants of the Vertex Cover problem (among others, Connected Vertex Cover, Capacitated Vertex Cover, and Maximum
MSP Algorithm: MultiRobot Patrolling based on Territory Allocation using Balanced Graph Partitioning
"... This article addresses the problem of efficient multirobot patrolling in a known environment. The proposed approach assigns regions to each mobile agent. Every region is represented by a subgraph extracted from the topological representation of the global environment. A new algorithm is proposed in ..."
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This article addresses the problem of efficient multirobot patrolling in a known environment. The proposed approach assigns regions to each mobile agent. Every region is represented by a subgraph extracted from the topological representation of the global environment. A new algorithm is proposed in order to deal with the local patrolling task assigned for each robot, named Multilevel Subgraph Patrolling (MSP) Algorithm. It handles some major graph theory classic problems like graph partitioning, Hamilton cycles, nonHamilton cycles and longest path searches. The flexible, scalable, robust and high performance nature of this approach is testified by simulation results.
Efficient GoalDirected Exploration
 IN PROCEEDINGS OF THE NATIONAL CONFERENCE ON AI
, 1996
"... If a state space is not completely known in advance, then search algorithms have to explore it sufficiently to locate a goal state and a path leading to it, performing therefore what we call goaldirected exploration. Two paradigms of this process are pure exploration and heuristicdriven exploi ..."
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Cited by 9 (6 self)
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If a state space is not completely known in advance, then search algorithms have to explore it sufficiently to locate a goal state and a path leading to it, performing therefore what we call goaldirected exploration. Two paradigms of this process are pure exploration and heuristicdriven exploitation: the former approaches explore the state space using only knowledge of the physically visited portion of the domain, whereas the latter approaches totally rely on heuristic knowledge to guide the search towards goal states. Both approaches have disadvantages: the first one does not utilize available knowledge to cut down the search effort, and the second one relies too much on the knowledge, even if it is misleading. We have therefore developed a framework for goaldirected exploration, called VECA, that combines the advantages of both approaches by automatically switching from exploitation to exploration on parts of the state space where exploitation does not perform well. VECA provides better performance guarantees than previously studied heuristicdriven exploitation algorithms, and experimental evidence suggests that this guarantee does not deteriorate its averagecase performance.
2008a), “Pairwise Display of highdimensional information via Eulerian tours and Hamiltonian decompositions
, 1975
"... A graph theoretic approach is taken to the component order problem in the layout of statistical graphics. Eulerian tours and Hamiltonian decompositions of complete graphs are used to ameliorate order effects in statistical graphics. Similar traversals of edge weighted graphs are used to amplify the ..."
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Cited by 6 (1 self)
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A graph theoretic approach is taken to the component order problem in the layout of statistical graphics. Eulerian tours and Hamiltonian decompositions of complete graphs are used to ameliorate order effects in statistical graphics. Similar traversals of edge weighted graphs are used to amplify the visual effect of selected salient features in the data. Relevant graph theory is summarized and classic algorithms are tailored to this problem. Graphics for multiple comparisons are reviewed and a new display developed that is based on graph traversal. Interaction plots are improved and new ones proposed. Improved star glyph displays of multivariate data are described. Parallel coordinate displays tailored to particular features of the data are developed. The methods and new graphical displays are made available as an R package. 1
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