The problem of enumerating connected induced subgraphs of a given graph is classical and studied well. It is known that connected induced subgraphs can be enumerated in constant time for each subgraph. In this paper, we focus on highly connected induced subgraphs. The most major concept of connectivity on graphs is vertex connectivity. For vertex connectivity, some enumeration problem settings and enumeration algorithms have been proposed, such as k-vertex connected spanning subgraphs. In this paper, we focus on another major concept of graph connectivity, edge-connectivity. This is motivated by the problem of finding evacuation routes in road networks. In evacuation routes, edge-connectivity is important, since highly edge-connected subgraphs ensure multiple routes between two vertices. In this paper, we consider the problem of enumerating 2-edge-connected induced subgraphs of a given graph. We present an algorithm that enumerates 2-edge-connected induced subgraphs of an input graph G with n vertices and m edges. Our algorithm enumerates all the 2-edge-connected induced subgraphs in O(n3m|SG|) time, where SG is the set of the 2-edge-connected induced subgraphs of G. Moreover, by slightly modifying the algorithm, we have a O(n3m)-delay enumeration algorithm for 2-edge-connected induced subgraphs.

Ising machines have attracted attention as they are expected to solve combinatorial optimization problems at high speed with Ising models corresponding to those problems. An induced subgraph isomorphism problem is one of the decision problems, which determines whether a specific graph structure is included in a whole graph or not. The problem can be represented by equality constraints in the words of combinatorial optimization problem. By using the penalty functions corresponding to the equality constraints, we can utilize an Ising machine to the induced subgraph isomorphism problem. The induced subgraph isomorphism problem can be seen in many practical problems, for example, finding out a particular malicious circuit in a device or particular network structure of chemical bonds in a compound. However, due to the limitation of the number of spin variables in the current Ising machines, reducing the number of spin variables is a major concern. Here, we propose an efficient Ising model mapping method to solve the induced subgraph isomorphism problem by Ising machines. Our proposed method theoretically solves the induced subgraph isomorphism problem. Furthermore, the number of spin variables in the Ising model generated by our proposed method is theoretically smaller than that of the conventional method. Experimental results demonstrate that our proposed method can successfully solve the induced subgraph isomorphism problem by using the Ising-model based simulated annealing and a real Ising machine.

This paper deals with the problem of enumerating 3-edge-connected spanning subgraphs of an input plane graph. In 2018, Yamanaka et al. proposed two enumeration algorithms for such a problem. Their algorithm generates each 2-edge-connected spanning subgraph of a given plane graph with n vertices in O(n) time, and another one generates each k-edge-connected spanning subgraph of a general graph with m edges in O(mT) time, where T is the running time to check the k-edge connectivity of a graph. This paper focuses on the case of the 3-edge-connectivity in a plane graph. We give an algorithm which generates each 3-edge-connected spanning subgraph of the input plane graph in O(n2) time. This time complexity is the same as the algorithm by Yamanaka et al., but our algorithm is simpler than theirs.

This paper proposes a method for searching for graphs in the database which are contained as subgraphs by a given query. In the proposed method, the search index does not require any knowledge of the query set or the frequent subgraph patterns. In conventional techniques, enumerating and selecting frequent subgraph patterns is computationally expensive, and the distribution of the query set must be known in advance. Subsequent changes to the query set require the frequent patterns to be selected again and the index to be reconstructed. The proposed method overcomes these difficulties through graph coding, using a tree structured index that contains infrequent subgraph patterns in the shallow part of the tree. By traversing this code tree, we are able to rapidly determine whether multiple graphs in the database contain subgraphs that match the query, producing a powerful pruning or filtering effect. Furthermore, the filtering and verification steps of the graph search can be conducted concurrently, rather than requiring separate algorithms. As the proposed method does not require the frequent subgraph patterns and the query set, it is significantly faster than previous techniques; this independence from the query set also means that there is no need to reconstruct the search index when the query set changes. A series of experiments using a real-world dataset demonstrate the efficiency of the proposed method, achieving a search speed several orders of magnitude faster than the previous best.

In this paper, we consider the problem of enumerating spanning subgraphs with high edge-connectivity of an input graph. Such subgraphs ensure multiple routes between two vertices. We first present an algorithm that enumerates all the 2-edge-connected spanning subgraphs of a given plane graph with n vertices. The algorithm generates each 2-edge-connected spanning subgraph of the input graph in O(n) time. We next present an algorithm that enumerates all the k-edge-connected spanning subgraphs of a given general graph with m edges. The algorithm generates each k-edge-connected spanning subgraph of the input graph in O(mT) time, where T is the running time to check the k-edge-connectivity of a graph.

For a family H of connected graphs and an integer k≥1, let Gk(H) denote the family of k-connected graphs which contain no element of H as an induced subgraph. Let H+ be the family of those connected graphs of order 5 which contain K1,3 as an induced subgraph. In this paper, for each integer k≥1, we characterize the families H⊆H+ such that the symmetric difference of Gk(K1,3) and Gk(H) is finite.

In this study, we address a problem pertaining to the induced matching enumeration. An edge set M is an induced matching of a graph G=(V,E). The enumeration of matchings has been widely studied in literature; however, there few studies on induced matching. A straightforward algorithm takes O(Δ2) time for each solution that is coming from the time to generate a subproblem, where Δ is the maximum degree in an input graph. To generate a subproblem, an algorithm picks up an edge e and generates two graphs, the one is obtained by removing e from G, the other is obtained by removing e, adjacent edge to e, and edges adjacent to adjacent edge of e. Since this operation needs O(Δ2) time, a straightforward algorithm enumerates all induced matchings in O(Δ2) time per solution. We investigated local structures that enable us to generate subproblems within a short time and proved that the time complexity will be O(1) if the input graph is C4-free. A graph is C4-free if and only if none of its subgraphs have a cycle of length four.

This paper proposes a regulated transport network design algorithm for IP over a dense wavelength division multiplex (DWDM) network. When designing an IP over DWDM network, the network operator should consider not only cost-effectiveness and physical constraints such as wavelength colors and chromatic dispersion but also operational policies such as resilience, quality, stability, and operability. For considering the above polices, we propose to separate the network design algorithm based on a geographical resolution; the policy-based regulated intra-area is designed based on this resolution, and the cost-optimal inter-area is then designed separately, and finally merged. This approach does not necessarily yield a strict optimal solution, but it covers network design work done by humans, which takes a vast amount of time and requires a high skill level. For efficient geographical resolution, we also present fast graph mining algorithm, which can solve NP-hard subgraph isomorphism problem within the practical time. We prove the sufficiency of the resulting network design for the above polices by visualizing the topology, and also prove that the penalty of applying the approach is trivial.

For subgraph enumeration problems, very efficient algorithms have been proposed whose time complexities are far smaller than the number of subgraphs. Although the number of subgraphs can exponentially increase with the input graph size, these algorithms exploit compressed representations to output and maintain enumerated subgraphs compactly so as to reduce the time and space complexities. However, they are designed for enumerating only some specific types of subgraphs, e.g., paths or trees. In this paper, we propose an algorithm framework, called the frontier-based search, which generalizes these specific algorithms without losing their efficiency. Our frontier-based search will be used to resolve various practical problems that include constrained subgraph enumeration.

Subgraph matching is a fundamental operation for querying graph-structured data. Due to potential errors and noises in real-world graph data, exact subgraph matching is sometimes inappropriate in practice. In this paper we consider an approximate subgraph matching model that allows missing edges. Based on this model, approximate subgraph matching finds all occurrences of a given query graph in a database graph, allowing missing edges. A straightforward approach is to first generate query subgraphs of a given query graph by deleting edges and then perform exact subgraph matching for each query subgraph. In this paper we propose a sharing-based approach to approximate subgraph matching, called SASUM. Our method is based on the fact that query subgraphs are highly overlapped. Due to this overlapping nature of query subgraphs, the matches of a query subgraph can be computed from the matches of a smaller query subgraph, which results in reducing the number of query subgraphs that require expensive exact subgraph matching. Our method uses a lattice framework to identify sharing opportunities between query subgraphs. To further reduce the number of graphs that need exact subgraph matching, SASUM generates small base graphs that are shared by query subgraphs and chooses the minimum number of base graphs whose matches are used to derive the matching results of all query subgraphs. A comprehensive set of experiments shows that our approach outperforms the state-of-the-art approach by orders of magnitude in terms of query execution time.

Given a connected graph G = (V, E) on n vertices, the MAXIMUM r-REGULAR INDUCED CONNECTED SUBGRAPH (r-MaxRICS) problem asks for a maximum sized subset of vertices S ⊆ V such that the induced subgraph G[S] on S is connected and r-regular. It is known that 2-MaxRICS and 3-MaxRICS are NP-hard. Moreover, 2-MaxRICS cannot be approximated within a factor of n1-ε in polynomial time for any ε > 0 unless P= NP. In this paper, we show that r-MaxRICS are NP-hard for any fixed integer r ≥ 4. Furthermore, we show that for any fixed integer r ≥ 3, r-MaxRICS cannot be approximated within a factor of n1/6-ε in polynomial time for any ε > 0 unless P= NP.

The problem of deciding whether a graph contains another graph appears in various applications. For solving this problem efficiently, we developed a numerical method to detect non-subgraphs, graphs which are not subgraphs of other graphs, by comparing eigenvalues of graphs. In this paper, we propose a method to make the detection more efficient by comparing of eigenvalues of graphs decomposed according to labels of the vertices and the edges. The new approach not only reduces the cost of computing eigenvalues but also increases the possibility of detecting non-subgraphs. The experimental evaluation shows the effectiveness of the proposed method.

The mining of a complete set of frequent subgraphs from labeled graph data has been studied extensively. Furthermore, much attention has recently been paid to frequent pattern mining from graph sequences (dynamic graphs or evolving graphs). In this paper, we define a novel subgraph subsequence class called an “induced subgraph subsequence” to enable the efficient mining of a complete set of frequent patterns from graph sequences containing large graphs and long sequences. We also propose an efficient method for mining frequent patterns, called “FRISSs (Frequent Relevant, and Induced Subgraph Subsequences)”, from graph sequences. The fundamental performance of the method is evaluated using artificial datasets, and its practicality is confirmed through experiments using a real-world dataset.

Let Ni be the number of connected spanning subgraphs with i(n-1 i m) edges in an n-vertex m-edge undirected graph G=(V,E). Although Nn-1 is computed in polynomial time by the Matrix-tree theorem, whether Nn is efficiently computed for a graph G is an open problem (see e.g., [2]). On the other hand, whether Nn2≥ Nn-1Nn+1 for a graph G is also open as a part of log concave conjecture (see e.g., [6],[12]). In this paper, for a complete graph Kn, we give the formulas for Nn, Nn+1, by which Nn, Nn+1 are respectively computed in polynomial time on n, and, in particular, prove Nn2> Nn-1Nn+1 as well.

Consider an undirected multigraph G=(V,E) with n vertices and m edges, and let Ni denote the number of connected spanning subgraphs with i(min) edges in G. Recently, we showed in [3] the validity of (m-i+1)Ni-1>Ni for a simple graph and each i(min). Note that, from this inequality, 2 is easily derived. In this paper, for a multigraph G and all i(min), we prove (m-i+1)Ni-1(i-n+2)Ni, and give a necessary and sufficient condition by which (m-i+1)Ni-1=(i-n+2)Ni. In particular, this means that (m-i+1)Ni-1>Ni is not valid for all multigraphs, in general. Furthermore, we prove 2, which is not straightforwardly derived from (m-i+1)Ni-1(i-n+2)Ni, and also introduce a necessary and sufficent condition by which =2. Moreover, we show a sufficient condition for a multigraph to have Nn2>Nn-1Nn+1. As special cases of the sufficient condition, we show that if G contains at least +1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then Nn2>Nn-1Nn+1.

In this paper, we present a hill-shift learning method of the Hopfield neural network for bipartite subgraph problem. The method uses the Hopfield neural network to get a near-maximum bipartite subgraph, and shifts the local minimum of energy function by adjusts the balance between two terms in the energy function to help the network escape from the state of the near-maximum bipartite subgraph to the state of the maximum bipartite subgraph or better one. A large number of instances are simulated to verify the proposed method with the simulation results showing that the solution quality is superior to that of best existing parallel algorithm.

We formally define the mobile agent allocation problem from a system-wide viewpoint and then prove that it is strongly NP-complete even if each agent communicates only with two agents. This is the first formal definition for scheduling mobile agents from the viewpoint of load balancing, which enables us to discuss its properties on a rigorous basis. The problem is recognized as preemptive scheduling with independent tasks that require mutual communication. The result implies that almost all subproblems of mobile agent allocation, which require mutual communication of agents, are strongly NP-complete.

Consider an undirected graph G=(V,E) with n (=|V|) vertices and m (=|E|) edges. It is well-known that the problem of computing the sequence Nn-1,Nn,...,Nm is #P-complete (see e.g.,[3]), where Ni denotes the number of connected spanning subgraphs with i (n-1!im) edges in G. In this paper, by proving new inequalities on the sequence Nn-1,Nn,...,Nm, we show an interesting and stronger property that the sequence γn-1,γn,...,γm, where γi denotes the average number of spanning trees in the connected spanning subgraphs with i edges, is a convex sequence as well as a monotonically increasing sequence, although this property does not hold for the sequence Nn-1,Nn,...,Nm.

A near-optimum parallel algorithm for bipartite subgraph problem using gradient ascent learning algorithm of the Hopfield neural networks is presented. This parallel algorithm, uses the Hopfield neural network updating to get a near-maximum bipartite subgraph and then performs gradient ascent learning on the Hopfield network to help the network escape from the state of the near-maximum bipartite subgraph until the state of the maximum bipartite subgraph or better one is obtained. A large number of instances have been simulated to verify the proposed algorithm, with the simulation result showing that our algorithm finds the solution quality is superior to that of best existing parallel algorithm. We also test the proposed algorithm on maximum cut problem. The simulation results also show the effectiveness of this algorithm.

Efficient content-based retrieval of complex images is a challenging task since the detected object may appear in various scale, rotation and orientation with a wide variety of background colors and forms. In this paper, we propose a novel representation of objects with multiple colors, the spatial neighborhood-adjacency graph(SNAG), which can serve as a basis for detecting object by color contents from the candidate image. The SNAG consists of a set of main-vertices and two sets of edges. Each main-vertex represents a single color region of multi-colored object, and edges are divided into two classes: Neighborhood edges representing neighborhood relationship between two main-vertices with similar color, and adjacency edges representing adjacency relationship between a main-vertex and another vertex with different color. By investigating whether SNAG of object image is an isomorphic subgraph of SNAG of a candidate image, we can determine whether the similar object exists in the candidate image. In addition, we have also applied the proposed approach to a range of different object detection problems involving complex background, and effectiveness has been proved.