84,485 research outputs found
Vertex-Magic Total Labeling on G-sun Graphs
Graph labeling is an immense area of research in mathematics, specifically graph theory. There are many types of graph labelings such as harmonious, magic, and lucky labelings. This paper will focus on magic labelings. Graph theorists are particularly interested in magic labelings because of a simple problem regarding tree graphs introduced in the 1990’s. The problem is still unsolved after almost thirty years. Researchers have studied magic labelings on other graphs in addition to tree graphs. In this paper we will consider vertex-magic labelings on G-sun graphs. We will give vertex-magic total labelings for ladder sun graphs and complete bipartite sun graphs. We will also show when there is no vertex-magic total labeling for other types of G-sun graphs
The Complexity of Routing with Few Collisions
We study the computational complexity of routing multiple objects through a
network in such a way that only few collisions occur: Given a graph with
two distinct terminal vertices and two positive integers and , the
question is whether one can connect the terminals by at least routes (e.g.
paths) such that at most edges are time-wise shared among them. We study
three types of routes: traverse each vertex at most once (paths), each edge at
most once (trails), or no such restrictions (walks). We prove that for paths
and trails the problem is NP-complete on undirected and directed graphs even if
is constant or the maximum vertex degree in the input graph is constant.
For walks, however, it is solvable in polynomial time on undirected graphs for
arbitrary and on directed graphs if is constant. We additionally study
for all route types a variant of the problem where the maximum length of a
route is restricted by some given upper bound. We prove that this
length-restricted variant has the same complexity classification with respect
to paths and trails, but for walks it becomes NP-complete on undirected graphs
Assessing the Computational Complexity of Multi-Layer Subgraph Detection
Multi-layer graphs consist of several graphs (layers) over the same vertex
set. They are motivated by real-world problems where entities (vertices) are
associated via multiple types of relationships (edges in different layers). We
chart the border of computational (in)tractability for the class of subgraph
detection problems on multi-layer graphs, including fundamental problems such
as maximum matching, finding certain clique relaxations (motivated by community
detection), or path problems. Mostly encountering hardness results, sometimes
even for two or three layers, we can also spot some islands of tractability
The -matching problem on bipartite graphs
The -matching problem on bipartite graphs is studied with a local
algorithm. A -matching () on a bipartite graph is a set of matched
edges, in which each vertex of one type is adjacent to at most matched edge
and each vertex of the other type is adjacent to at most matched edges. The
-matching problem on a given bipartite graph concerns finding -matchings
with the maximum size. Our approach to this combinatorial optimization are of
two folds. From an algorithmic perspective, we adopt a local algorithm as a
linear approximate solver to find -matchings on general bipartite graphs,
whose basic component is a generalized version of the greedy leaf removal
procedure in graph theory. From an analytical perspective, in the case of
random bipartite graphs with the same size of two types of vertices, we develop
a mean-field theory for the percolation phenomenon underlying the local
algorithm, leading to a theoretical estimation of -matching sizes on
coreless graphs. We hope that our results can shed light on further study on
algorithms and computational complexity of the optimization problem.Comment: 15 pages, 3 figure
Beyond topological persistence: Starting from networks
Persistent homology enables fast and computable comparison of topological
objects. However, it is naturally limited to the analysis of topological
spaces. We extend the theory of persistence, by guaranteeing robustness and
computability to significant data types as simple graphs and quivers. We focus
on categorical persistence functions that allow us to study in full generality
strong kinds of connectedness such as clique communities, -vertex and
-edge connectedness directly on simple graphs and monic coherent categories.Comment: arXiv admin note: text overlap with arXiv:1707.0967
Route systems on graphs
summary:The well known types of routes in graphs and directed graphs, such as walks, trails, paths, and induced paths, are characterized using axioms on vertex sequences. Thus non-graphic characterizations of the various types of routes are obtained
First Passage Percolation on Inhomogeneous Random Graphs
We investigate first passage percolation on inhomogeneous random graphs. The
random graph model G(n,kappa) we study is the model introduced by Bollob\'as,
Janson and Riordan, where each vertex has a type from a type space S and edge
probabilities are independent, but depending on the types of the end vertices.
Each edge is given an independent exponential weight. We determine the
distribution of the weight of the shortest path between uniformly chosen
vertices in the giant component and show that the hopcount, i.e. the number of
edges on this minimal weight path, properly normalized follows a central limit
theorem. We handle the cases where lambda(n)->lambda is finite or infinite,
under the assumption that the average number of neighbors lambda(n) of a vertex
is independent of the type. The paper is a generalization the paper by Bhamidi,
van der Hofstad and Hooghiemstra, where FPP is explored on the Erdos-Renyi
graphs
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