# Category: Computational problems in graph theory

Spanning tree
In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may have several spannin
Widest path problem
In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The wid
Strong connectivity augmentation
Strong connectivity augmentation is a computational problem in the mathematical study of graph algorithms, in which the input is a directed graph and the goal of the problem is to add a small number o
Minimum k-cut
In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges
MaxDDBS
The Maximum Degree-and-Diameter-Bounded Subgraph problem (MaxDDBS) is a problem in graph theory. Given a connected host graph G, an upper bound for the degree d, and an upper bound for the diameter k,
Digraph realization problem
The digraph realization problem is a decision problem in graph theory. Given pairs of nonnegative integers , the problem asks whether there is a labeled simple directed graph such that each vertex has
K shortest path routing
The k shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next k−1 shortest paths (which ma
Travelling salesman problem
The travelling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the
Good spanning tree
In the mathematical field of graph theory, a good spanning tree of an embedded planar graph is a rooted spanning tree of whose non-tree edges satisfy the following conditions. * there is no non-tree
Planted clique
In computational complexity theory, a planted clique or hidden clique in an undirected graph is a clique formed from another graph by selecting a subset of vertices and adding edges between each pair
Snake-in-the-box
The snake-in-the-box problem in graph theory and computer science deals with finding a certain kind of path along the edges of a hypercube. This path starts at one corner and travels along the edges t
Hamiltonian path problem
In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed gra
Degree diameter problem
In graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the largest degree of any of the
Maximum weight matching
In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is t
Hamiltonian path
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian
In computer science and graph theory, the Canadian traveller problem (CTP) is a generalization of the shortest path problem to graphs that are partially observable. In other words, the graph is reveal
Feedback vertex set
In the mathematical discipline of graph theory, a feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles ("removal" means deleting the vertex and all edg
Network simplex algorithm
In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. T
Hamiltonian cycle polynomial
In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as where is the set of n-permutations having exactly one cycle. This is an algebraic option us
Domatic number
In graph theory, a domatic partition of a graph is a partition of into disjoint sets , ,..., such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph;
Graph edit distance
In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs.The concept of graph edit distance was first formalized mathematically b
Longest path problem
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any r
Steiner tree problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Ste
Correlation clustering
Clustering is the problem of partitioning data points into groups based on their similarity. Correlation clustering provides a method for clustering a set of objects into the optimum number of cluster
Deterministic rendezvous problem
The deterministic rendezvous problem is a variant of the rendezvous problem where the players, or robots, must find each other by following a deterministic sequence of instructions. Although each robo
Graph realization problem
The graph realization problem is a decision problem in graph theory. Given a finite sequence of natural numbers, the problem asks whether there is a labeled simple graph such that is the degree sequen
Frequent subtree mining
In computer science, frequent subtree mining is the problem of finding all patterns in a given database whose support (a metric related to its number of occurrences in other subtrees) is over a given
Metric k-center
In graph theory, the metric k-center or metric facility location problem is a combinatorial optimization problem studied in theoretical computer science. Given n cities with specified distances, one w
Odd cycle transversal
In graph theory, an odd cycle transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd c
Wiener connector
In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a
In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphssuch that the labels of adjacent
Facility location problem
The study of facility location problems (FLP), also known as location analysis, is a branch of operations research and computational geometry concerned with the optimal placement of facilities to mini
Vertex cover
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a min
Dominating set
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is either in D, or has a neighbor in D. The domination number γ(G) is the number of vertices in
Induced subgraph isomorphism problem
In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph.
Vertex cycle cover
In mathematics, a vertex cycle cover (commonly called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G. If the cycles of the cover have no ver
Clique problem
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several dif
Maximum flow problem
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case
Nonblocker
In graph theory, a nonblocker is a subset of vertices in an undirected graph, all of which are adjacent to vertices outside of the subset. Equivalently, a nonblocker is the complement of a dominating
Maximal independent set
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independe
Mixed Chinese postman problem
The mixed Chinese postman problem (MCPP or MCP) is the search for the shortest traversal of a graph with a set of vertices V, a set of undirected edges E with positive rational weights, and a set of d
Connected dominating set
In graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph.
Chinese postman problem
In graph theory, a branch of mathematics and computer science, Guan's route problem, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed path or circuit
Maximum agreement subtree problem
The maximum agreement subtree problem is any of several closely related problems in graph theory and computer science. In all of these problems one is given a collection of trees each containing leave
Planarity testing
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections).
Subgraph isomorphism problem
In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G and H are given as input, and one must determine whether G contains a subgraph that is i
Pebble motion problems
The pebble motion problems, or pebble motion on graphs, are a set of related problems in graph theory dealing with the movement of multiple objects ("pebbles") from vertex to vertex in a graph with a
Seidel's algorithm
Seidel's algorithm is an algorithm designed by Raimund Seidel in 1992 for the all-pairs-shortest-path problem for undirected, unweighted, connected graphs. It solves the problem in expected time for a
Set TSP problem
In combinatorial optimization, the set TSP, also known as the generalized TSP, group TSP, One-of-a-Set TSP, Multiple Choice TSP or Covering Salesman Problem, is a generalization of the traveling sales
Graph sandwich problem
In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of whic
Bipartite realization problem
The bipartite realization problem is a classical decision problem in graph theory, a branch of combinatorics. Given two finite sequences and of natural numbers, the problem asks whether there is label
Clique cover
In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices into cliques, subsets of vertices within which every two vertices are adjacent. A m
Graph coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest
Graph cuts in computer vision
As applied in the field of computer vision, graph cut optimization can be employed to efficiently solve a wide variety of low-level computer vision problems (early vision), such as image smoothing, th
Graph matching
Graph matching is the problem of finding a similarity between graphs. Graphs are commonly used to encode structural information in many fields, including computer vision and pattern recognition, and g
Graph partition
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the gro
Multi-trials technique
The multi-trials technique by Schneider et al. is employed for distributed algorithms and allows breaking of symmetry efficiently. Symmetry breaking is necessary, for instance, in resource allocation
Independent set (graph theory)
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices
Token reconfiguration
In computational complexity theory and combinatorics, the token reconfiguration problem is a reconfiguration problem on a graph with both an initial and desired state for tokens. Given a graph , an in
Graph isomorphism problem
Unsolved problem in computer science: Can the graph isomorphism problem be solved in polynomial time? (more unsolved problems in computer science) The graph isomorphism problem is the computational pr
Instant Insanity
Instant Insanity is the name given by Parker Brothers to their 1967 version of a puzzle which has existed since antiquity, and which has been marketed by many toy and puzzle makers under a variety of
Maximum common induced subgraph
In graph theory and theoretical computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H,and that has as many vertices as possib
Maximum common edge subgraph
Given two graphs and , the maximum common edge subgraph problem is the problem of finding a graph with as many edges as possible which is isomorphic to both a subgraph of and a subgraph of . The maxim
Edge dominating set
In graph theory, an edge dominating set for a graph G = (V, E) is a subset D ⊆ E such that every edge not in D is adjacent to at least one edge in D. An edge dominating set is also known as a line dom
Feedback arc set
In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph.
Longest uncrossed knight's path
The longest uncrossed (or nonintersecting) knight's path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find t
Edge cover
In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set.In computer science, the minimum edge cover problem is the p
Matching (graph theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be trea
Graph cut optimization
Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the ma
Maximum cut
For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of ed
Shortest path problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The probl