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Register a new userOn finding orientations with fewest number of vartices with small out-degree
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Added by
Kaveh Khoshkhah
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Authors
Kaveh Khoshkhah
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Given an undirected graph, each of the two end-vertices of an edge can own the edge. Call a vertex poor, if it owns at most one edge. We give a polynomial time algorithm for the problem of finding an assignment of owners to the edges which minimizes the number of poor vertices. In the terminology of graph orientation, this means finding an orientation for the edges of a graph minimizing the number of edges with out-degree at most 1, and answers a question of Asahiro Jansson, Miyano, Ono (2014).
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The Rooted Maximum Leaf Outbranching problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least $k$ leaves. We use the notion of $s-t$ numbering to exhibit combinatorial bounds on the existence of spanning directed trees with many leaves. These combinatorial bounds allow us to produce a constant factor approximation algorithm for finding directed trees with many leaves, whereas the best known approximation algorithm has a $sqrt{OPT}$-factor. We also show that Rooted Maximum Leaf Outbranching admits a quadratic kernel, improving over the cubic kernel given by Fernau et al.
Given a digraph $D$ with $m $ arcs, a bijection $tau: A(D)rightarrow {1, 2, ldots, m}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. We say $(D, tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $tau$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, M{u}tze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph $G$ on $nge9$ vertices with maximum degree at least $n-5$ admits an antimagic orientation.
When we try to solve a system of linear equations, we can consider a simple iterative algorithm in which an equation including only one variable is chosen at each step, and the variable is fixed to the value satisfying the equation. The dynamics of this algorithm is captured by the peeling algorithm. Analyses of the peeling algorithm on random hypergraphs are required for many problems, e.g., the decoding threshold of low-density parity check codes, the inverting threshold of Goldreichs pseudorandom generator, the load threshold of cuckoo hashing, etc. In this work, we deal with random hypergraphs including superlinear number of hyperedges, and derive the tight threshold for the succeeding of the peeling algorithm. For the analysis, Wormalds method of differential equations, which is commonly used for analyses of the peeling algorithm on random hypergraph with linear number of hyperedges, cannot be used due to the superlinear number of hyperedges. A new method called the evolution of the moment generating function is proposed in this work.
Many real-world networks exhibit correlations between the node degrees. For instance, in social networks nodes tend to connect to nodes of similar degree. Conversely, in biological and technological networks, high-degree nodes tend to be linked with low-degree nodes. Degree correlations also affect the dynamics of processes supported by a network structure, such as the spread of opinions or epidemics. The proper modelling of these systems, i.e., without uncontrolled biases, requires the sampling of networks with a specified set of constraints. We present a solution to the sampling problem when the constraints imposed are the degree correlations. In particular, we develop an efficient and exact method to construct and sample graphs with a specified joint-degree matrix, which is a matrix providing the number of edges between all the sets of nodes of a given degree, for all degrees, thus completely specifying all pairwise degree correlations, and additionally, the degree sequence itself. Our algorithm always produces independent samples without backtracking. The complexity of the graph construction algorithm is O(NM) where N is the number of nodes and M is the number of edges.
Given a digraph $D$ with $m$ arcs and a bijection $tau: A(D)rightarrow {1, 2, ldots, m}$, we say $(D, tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $tau$, where the vertex-sum of a vertex $u$ in $D$ under $tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, M{u}tze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs, biregular bipartite graphs, and graphs with large maximum degree. In this paper, we establish more evidence for the aforementioned conjecture by studying antimagic orientations of graphs $G$ with independence number at least $|V(G)|/2$ or at most four. We obtain several results. The method we develop in this paper may shed some light on attacking the aforementioned conjecture.
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