Kunihiro Wasa

Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph

In this paper, we address the problem of enumerating all induced subtrees in an input \(k\) -degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph \(G = (V, E)\) is a \(k\)-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to \(k\), and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced subgraph versions have not been studied well. One of few examples is for chordless paths and cycles. Our motivation is to reduce the time complexity close to \(O(1)\) for each solution. This type of optimal algorithms is proposed many subgraph classes such as trees, and spanning trees. Induced subtrees are fundamental object thus it should be studied deeply and there possibly exist some efficient algorithms. Our algorithm utilizes nice properties of \(k\)-degeneracy to state an effective amortized analysis. As a result, the time complexity is reduced to \(O(k)\) time per induced subtree. The problem is solved in constant time for each in planar graphs, as a corollary.

Listing Acyclic Subgraphs and Subgraphs of Bounded Girth in Directed Graphs

The girth of a directed graph is the length of its shortest directed cycle. We consider the problem of generating all subgraphs of girth at least g in a directed graph G with n vertices and m edges. This generalizes the problem of generating acyclic subgraphs (i.e., with no directed cycle), that correspond to the subgraphs of girth at least \(n+1\). The problem of finding the acyclic subgraph with maximum size or weight has been thoroughly studied, however to the best of our knowledge there is no known efficient enumeration algorithm. We propose polynomial delay algorithms for listing both induced and edge subgraphs with girth g in time O(n) per solution; both improve upon a naive solution, respectively by a factor O(nm) and \(O(m^2)\). Furthermore, this work is on the line of existing research for extracting acyclic structures from graphs.

Efficient Enumeration of Maximal k-Degenerate Subgraphs in a Chordal Graph

In this paper, we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay \(O(m\cdot \omega (G))\) for any n-vertex chordal graph with m edges, where \(\omega (G) \le n\) is the maximum size of a clique in G. The problem generalizes that of enumerating maximal independent sets and maximal induced forests, which correspond to respectively 0-degenerate and 1-degenerate subgraphs.

The Complexity of Induced Tree Reconfiguration Problems

A reconfiguration problem asks when we are given two feasible solutions A and B, whether there exists a reconfiguration sequence \((A_0 = A, A_1, \dots , A_\ell = B)\) such that (i) \(A_0, \dots , A_\ell \) are feasible solutions and (ii) we can obtain \(A_i\) from \(A_{i-1}\) under the prescribed rule (the reconfiguration rule) for each \(i = 1, \dots , \ell \). In this paper, we address the reconfiguration problem for induced trees, where an induced tree is a connected and acyclic induced graph of an input graph. This paper treats the following two rules as the prescribed rules: Token Jumping; removing u from an induced tree and adding v to the tree, and Token Sliding; removing u from an induced tree and adding v adjacent to u to the tree, where u and v are vertices in an input graph. As the main results, we show (I) the reconfiguration problem is PSPACE-complete, (II) the reconfiguration problem is W[1]-hard when parameterized by both the size of induced trees and the length of the reconfiguration sequence, and (III) there exists an FPT algorithm when parameterized by both the size of induced trees and the maximum degree of an input graph, under each of Token Jumping and Token Sliding.

Efficient Enumeration of Subgraphs and Induced Subgraphs with Bounded Girth

The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and recent literature. In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph. We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in O(n) amortized time per solution and using \(O(n^3)\) space. Furthermore, the algorithms can be easily adapted to relax the connectivity requirement and to deal with weighted graphs. As a byproduct, the second algorithm can be used to answer the well known question of finding the densest n-vertex graph(s) of girth k.

Efficient Enumeration of Bipartite Subgraphs in Graphs

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enumeration problem of bipartite subgraphs. Even though bipartite graphs are quite fundamental and have numerous applications in both theory and application, its enumeration algorithms have not been intensively studied, to the best of our knowledge. We propose the first non-trivial algorithms for enumerating all bipartite subgraphs in a given graph. As the main results, we develop two efficient algorithms: the one enumerates all bipartite induced subgraphs of a graph with degeneracy

Exact Algorithms for the Max-Min Dispersion Problem

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Faster Algorithms for Tree Similarity Based on Compressed Enumeration of Bounded-Sized Ordered Subtrees

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Polynomial Delay and Space Discovery of Connected and Acyclic Sub-Hypergraphs in a Hypergraph

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Enumeration of Enumeration Algorithms.

time per solution. The other enumerates all bipartite subgraphs in