# Induced path

Last updated

In the mathematical area of graph theory, an induced path in an undirected graph G is a path that is an induced subgraph of G. That is, it is a sequence of vertices in G such that each two adjacent vertices in the sequence are connected by an edge in G, and each two nonadjacent vertices in the sequence are not connected by any edge in G. An induced path is sometimes called a snake, and the problem of finding long induced paths in hypercube graphs is known as the snake-in-the-box problem.

## Contents

Similarly, an induced cycle is a cycle that is an induced subgraph of G; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes. An antihole is a hole in the complement of G, i.e., an antihole is a complement of a hole.

The length of the longest induced path in a graph has sometimes been called the detour number of the graph; [1] for sparse graphs, having bounded detour number is equivalent to having bounded tree-depth. [2] The induced path number of a graph G is the smallest number of induced paths into which the vertices of the graph may be partitioned, [3] and the closely related path cover number of G is the smallest number of induced paths that together include all vertices of G. [4] The girth of a graph is the length of its shortest cycle, but this cycle must be an induced cycle as any chord could be used to produce a shorter cycle; for similar reasons the odd girth of a graph is also the length of its shortest odd induced cycle.

## Example

The illustration shows a cube, a graph with eight vertices and twelve edges, and an induced path of length four in this graph. A straightforward case analysis shows that there can be no longer induced path in the cube, although it has an induced cycle of length six. The problem of finding the longest induced path or cycle in a hypercube, first posed by Kautz (1958), is known as the snake-in-the-box problem, and it has been studied extensively due to its applications in coding theory and engineering.

## Characterization of graph families

Many important graph families can be characterized in terms of the induced paths or cycles of the graphs in the family.

• Trivially, the connected graphs with no induced path of length two are the complete graphs, and the connected graphs with no induced cycle are the trees.
• A triangle-free graph is a graph with no induced cycle of length three.
• The cographs are exactly the graphs with no induced path of length three.
• The chordal graphs are the graphs with no induced cycle of length four or more.
• The even-hole-free graphs are the graphs in containing no induced cycles with an even number of vertices.
• The trivially perfect graphs are the graphs that have neither an induced path of length three nor an induced cycle of length four.
• By the strong perfect graph theorem, the perfect graphs are the graphs with no odd hole and no odd antihole.
• The distance-hereditary graphs are the graphs in which every induced path is a shortest path, and the graphs in which every two induced paths between the same two vertices have the same length.
• The block graphs are the graphs in which there is at most one induced path between any two vertices, and the connected block graphs are the graphs in which there is exactly one induced path between every two vertices.

## Algorithms and complexity

It is NP-complete to determine, for a graph G and parameter k, whether the graph has an induced path of length at least k. Garey & Johnson (1979) credit this result to an unpublished communication of Mihalis Yannakakis. However, this problem can be solved in polynomial time for certain graph families, such as asteroidal-triple-free graphs [5] or graphs with no long holes. [6]

It is also NP-complete to determine whether the vertices of a graph can be partitioned into two induced paths, or two induced cycles. [7] As a consequence, determining the induced path number of a graph is NP-hard.

The complexity of approximating the longest induced path or cycle problems can be related to that of finding large independent sets in graphs, by the following reduction. [8] From any graph G with n vertices, form another graph H with twice as many vertices as G, by adding to Gn(n  1)/2 vertices having two neighbors each, one for each pair of vertices in G. Then if G has an independent set of size k, H must have an induced path and an induced cycle of length 2k, formed by alternating vertices of the independent set in G with vertices of I. Conversely, if H has an induced path or cycle of length k, any maximal set of nonadjacent vertices in G from this path or cycle forms an independent set in G of size at least k/3. Thus, the size of the maximum independent set in G is within a constant factor of the size of the longest induced path and the longest induced cycle in H. Therefore, by the results of Håstad (1996) on inapproximability of independent sets, unless NP=ZPP, there does not exist a polynomial time algorithm for approximating the longest induced path or the longest induced cycle to within a factor of O(n1/2-ε) of the optimal solution.

Holes (and antiholes in graphs without chordless cycles of length 5) in a graph with n vertices and m edges may be detected in time (n+m2). [9]

## Atomic cycles

Atomic cycles are a generalization of chordless cycles, that contain no n-chords. Given some cycle, an n-chord is defined as a path of length n connecting two points on the cycle, where n is less than the length of the shortest path on the cycle connecting those points. If a cycle has no n-chords, it is called an atomic cycle, because it cannot be decomposed into smaller cycles. [10] In the worst case, the atomic cycles in a graph can be enumerated in O(m2) time, where m is the number of edges in the graph.

## Related Research Articles

In graph theory, the shortest path problem is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized.

Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.

In graph theory, a cycle in a graph is a non-empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices.

The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. The algorithm was first proposed by Alfonso Shimbel (1955), but is instead named after Richard Bellman and Lester Ford Jr., who published it in 1958 and 1956, respectively. Edward F. Moore also published the same algorithm in 1957, and for this reason it is also sometimes called the Bellman–Ford–Moore algorithm.

This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

In graph theory, an Eulerian trail is a trail in a finite graph that visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this:

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 Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals. Further well-known variants are the Euclidean Steiner tree problem and the rectilinear minimum Steiner tree problem.

In the mathematical discipline of graph theory, a vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. The problem of finding a minimum vertex cover is a classical optimization problem in computer science and is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory.

In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if for all we have .

In graph theory, the perfect graph theorem of László Lovász states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by Berge, and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs.

In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs or triangulated graphs.

In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset.

In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.

In graph theory, a branch of discrete mathematics, a distance-hereditary graph is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.

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 repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.

In the mathematical area of graph theory, an undirected graph G is strongly chordal if it is a chordal graph and every cycle of even length in G has an odd chord, i.e., an edge that connects two vertices that are an odd distance (>1) apart from each other in the cycle.

In graph theory, the graph bandwidth problem is to label the n vertices vi of a graph G with distinct integers f(vi) so that the quantity is minimized . The problem may be visualized as placing the vertices of a graph at distinct integer points along the x-axis so that the length of the longest edge is minimized. Such placement is called linear graph arrangement, linear graph layout or linear graph placement.

In graph theory, a branch of combinatorial mathematics, a block graph or clique tree is a type of undirected graph in which every biconnected component (block) is a clique.

In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices such that either the number of edges that the path passes through or the sum of the weights of the edges is minimized. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.

In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals. Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.

## References

• Barioli, Francesco; Fallat, Shaun; Hogben, Leslie (2004). "Computation of minimal rank and path cover number for certain graphs" (PDF). Linear Algebra Appl. 392: 289–303. doi:10.1016/j.laa.2004.06.019.CS1 maint: ref=harv (link)
• Berman, Piotr; Schnitger, Georg (1992). "On the complexity of approximating the independent set problem". Information and Computation. 96 (1): 77–94. doi:.CS1 maint: ref=harv (link)
• Buckley, Fred; Harary, Frank (1988). "On longest induced paths in graphs". Chinese Quart. J. Math. 3 (3): 61–65.CS1 maint: ref=harv (link)
• Chartrand, Gary; McCanna, Joseph; Sherwani, Naveed; Hossain, Moazzem; Hashmi, Jahangir (1994). "The induced path number of bipartite graphs". Ars Combin. 37: 191–208.CS1 maint: ref=harv (link)
• Garey, Michael R.; Johnson, David S. (1979). . W. H. Freeman. p.  196.CS1 maint: ref=harv (link)
• Gashler, Michael; Martinez, Tony (2012). "Robust manifold learning with CycleCut" (PDF). Connection Science. 24 (1): 57–69. doi:10.1080/09540091.2012.664122.CS1 maint: ref=harv (link)
• Gavril, Fănică (2002). "Algorithms for maximum weight induced paths". Information Processing Letters. 81 (4): 203–208. doi:10.1016/S0020-0190(01)00222-8.CS1 maint: ref=harv (link)
• Håstad, Johan (1996). "Clique is hard to approximate within n1−ε". Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science. pp. 627–636. doi:10.1109/SFCS.1996.548522.CS1 maint: ref=harv (link)
• Kautz, W. H. (1958). "Unit-distance error-checking codes". IRE Transactions on Electronic Computers. 7: 177–180.CS1 maint: ref=harv (link)
• Kratsch, Dieter; Müller, Haiko; Todinca, Ioan (2003). "Feedback vertex set and longest induced path on AT-free graphs". Graph-theoretic concepts in computer science. Berlin: Lecture Notes in Computer Science, Vol. 2880, Springer-Verlag. pp. 309–321. doi:10.1007/b93953. Archived from the original on 2006-11-25.CS1 maint: ref=harv (link)
• Le, Hoàng-Oanh; Le, Van Bang; Müller, Haiko (2003). "Splitting a graph into disjoint induced paths or cycles" (PDF). Discrete Appl. Math. 131 (1): 199–212. doi:10.1016/S0166-218X(02)00425-0. Archived from the original (The Second International Colloquium "Journées de l'Informatique Messine", Metz, 2000) on 2016-03-03.CS1 maint: ref=harv (link)
• Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012). "Chapter 6. Bounded height trees and tree-depth". Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics. 28. Heidelberg: Springer. pp. 115–144. doi:10.1007/978-3-642-27875-4. ISBN   978-3-642-27874-7. MR   2920058.CS1 maint: ref=harv (link)
• Nikolopoulos, Stavros D.; Palios, Leonidas (2004). "Hole and antihole detection in graphs". Proc. 15th ACM-SIAM Symposium on Discrete Algorithms. pp. 850–859.CS1 maint: ref=harv (link)