Path Cover Number Research Articles

Zero forcing, graphs on k parallel paths, and linear preservers

The zero forcing number of a simple loopless undirected graph, being an upper bound on the path cover number and the maximum nullity of the graph, is an important parameter in the study of the minimum rank problem. In this article, we show that the minimum k for which a graph G is a graph on k parallel paths is an upper bound on the zero forcing number of G, and hence an upper bound on the path number and maximum nullity of G. We also determine an upper bound on the possible size (number of edges) of a graph on k parallel paths. Finally we show that the only linear operators that preserve the zero forcing number of a graph are the vertex permutations.

On the isometric path partition problem

The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (partition) number of $(r\times s)$-dimensional grid is $s$ when $r \geq s(s-1)$. We establish that the isometric path cover (partition) number of $(r\times r)$-dimensional torus is $r$ when $r$ is even and is either $r$ or $r+1$ when $r$ is odd. Then, we demonstrate that the isometric path cover (partition) number of an $r$-dimensional Benes network is $2^r$. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids.

On the Path Cover Number of Connected Quasi-Claw-Free Graphs

Detecting vertex disjoint paths is one of the central issues in designing and evaluating an interconnection network. It is naturally related to routing among nodes and fault tolerance of the network. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and a path cover number of G denoted by p(G) = min{|P| : P is a path cover of G}. In this paper, we show that if the minimum degree sum of an independent set with k + 1 vertices in a connected quasi-claw-free graph G of order n is no less than n - k, then p(G) ≤ k - 1, where k ≥ 2. Examples illustrate that the degree sum condition in our result is sharp.

On the Relationship Between the Zero Forcing Number and Path Cover Number for Some Graphs

The zero forcing number of a graph is the minimum size of a zero forcing set. This parameter bounds the path cover number which is the minimum number of vertex-disjoint-induced paths that cover all the vertices of graph. In this paper, we investigate these two parameters and present an infinite number of graphs with the large difference between these two parameters and an infinite number of graphs with no difference between these parameters. Also, we prove a conjecture about the relationship between these parameters for some graphs.

Minimum Path Cover in Quasi-claw-Free Graphs

A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths; a minimum path cover of G is a path cover of G consisting of minimum number of paths; a path cover number of G, denoted by p(G), is the number of paths in a minimum path cover of G, i.e., $$p(G)=\min \{|{\mathcal {P}}|:$$$${\mathcal {P}}$$ is a path cover of $$G\}.$$ A graph G is quasi-claw-free if for any two vertices x, y with $$d(x,y)=2$$ in G, there exists a vertex u with $$u\in N(x)\cap N(y)$$ and $$N(u)\subseteq N[x]\cup N[y].$$ In this paper, we prove that for any quasi-claw-free graph G of order n, if $$\sigma _{k+1}(G)\ge {n-k}$$ for a positive integer k, then $$p(G)\le k,$$ where $$\sigma _{k+1}(G)$$ is the minimum degree sum of an independent set with $$k+1$$ vertices in G. Our result is a generalization of Ore’s sufficient conditions of hamiltonicity for quasi-claw-free graphs.

Nullity of a graph in terms of path cover number

ABSTRACT The nullity of a graph G, written as , is defined to be the multiplicity of zero eigenvalues of its adjacency matrix. A path cover of G is a set of vertex-disjoint induced paths in G such that every vertex of G is a vertex of one of the paths. The path cover number of G, written as , is the minimum size of a path cover of G. In this article, we prove that for a connected graph G in which every block is a cycle or a clique. Furthermore, if every block of G has at least three vertices, then if and only if every block of G is a cycle of size a multiple of 4.

Matching, path covers, and total forcing sets

A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The minimum cardinality of a total forcing set in G is its total forcing number, denoted Ft(G). The path cover number of G, denoted pc(G), is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover, while the matching number of G, denoted α′(G), is the number of edges in a maximum matching of G. Let T be a tree of order at least two. We observe that pc(T) + 1 ≤ Ft(T) ≤ 2pc(T), and we prove that Ft(T) ≤ α′(T) + pc(T). Further, we characterize the extremal trees achieving equality in these bounds.

Eigenvariety of nonnegative symmetric weakly irreducible tensors associated with spectral radius and its application to hypergraphs

For a nonnegative symmetric weakly irreducible tensor, its spectral radius is an eigenvalue of the tensor corresponding to a unique positive eigenvector called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety of the tensor associated with the spectral radius is the set of the eigenvectors of the tensor corresponding to the spectral radius considered in the complex projective space.In this paper we proved that such projective eigenvariety admits a module structure, which is determined by the support of the tensor and can be characterized explicitly by the Smith normal form of the incidence matrix of the tensor. We introduced two parameters: the stabilizing index and the stabilizing dimension of the tensor, where the former is exactly the cardinality of the projective eigenvariety and the latter is the composition length of the projective eigenvariety as a module. We give some upper bounds for the two parameters, and characterize the case that there is only one eigenvector of the tensor corresponding to the spectral radius, i.e. the Perron vector. By applying the above results to the adjacency tensor of a connected uniform hypergraph, we give some upper bounds for the two parameters in terms of the structural parameters of the hypergraph such as path cover number, matching number and the maximum length of paths.

Minimum rank and path cover number for generalized and double generalized cycle star graphs

For a given connected (undirected) graph G=(V,E), with V={1,…,n}, the minimum rank of G is defined to be the smallest possible rank over all symmetric matrices A=[aij] such that for i≠j, aij=0 if, and only if, {i,j}∉E. The path cover number of G is the minimum number of vertex-disjoint paths occurring as induced subgraphs of G that cover all the vertices of G. When G is a tree, the values of the minimum rank and of the path cover number are known as well the relationship between them. We study these values and their relationship for all graphs that have at most two vertices of degree greater than two: generalized cycle stars and double generalized cycle stars.

Covering 2‐connected 3‐regular graphs with disjoint paths

AbstractA path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number of graph G is the cardinality of a path cover with the minimum number of paths. Reed in 1996 conjectured that a 2‐connected 3‐regular graph has path cover number at most . In this article, we confirm this conjecture.

Hydras: Directed hypergraphs and Horn formulas

We introduce a new graph parameter, the hydra number, arising from the minimization problem for Horn formulas in propositional logic. The hydra number of a graph G=(V,E) is the minimal number of hyperarcs of the form u,v→w required in a directed hypergraph H=(V,F), such that for every pair (u,v), the set of vertices reachable in H from {u,v} is the entire vertex set V if (u,v)∈E, and it is {u,v} otherwise. Here reachability is defined by forward chaining, a standard marking algorithm.Various bounds are given for the hydra number. We show that the hydra number of a graph can be upper bounded by the number of edges plus the path cover number of the line graph of a spanning subgraph, which is a sharp bound in several cases. On the other hand, we construct single-headed graphs for which that bound is off by a constant factor. Furthermore, we characterize trees with low hydra number, and give a lower bound for the hydra number of trees based on the number of vertices that are leaves in the tree obtained from T by deleting its leaves. This bound is sharp for some families of trees. We give bounds for the hydra number of complete binary trees and also discuss a related minimization problem.

Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs

∗Department of Mathematics, Statistics, and Computer Science, St. Olaf College, Northfield, MN 55057 (berliner@stolaf.edu), (coxn@stolaf.edu). Research of N. Cox supported by NSF DMS 0750986 †Department of Mathematics, Carleton College, Northfield, MN 55057 (brownc@carleton.edu). Research supported by NSF DMS 0750986. ‡Department of Mathematics, Iowa State University, Ames, IA 50011 (jmsdg7@iastate.edu), (warnberg@iastate.edu), (myoung@iastate.edu). Research of J. Carlson supported by ISU Holl funds. Research of M. Young supported by NSF DMS 0946431. §Department of Mathematics, Iowa State University, Ames, IA 50011, (lhogben@iastate.edu) and American Institute of Mathematics, 360 Portage Ave, Palo Alto, CA 94306 (hogben@aimath.org). ¶Department of Mathematics, University of California, Berkeley, CA 94720-3840 (jason.hu@berkeley.edu). Research supported by NSF DMS 0750986. ‖Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, CA 91711 (klj02011@mymail.pomona.edu). Research supported by NSF DMS 0750986. ∗∗Department of Mathematics and Computer Science, Central College, Pella, IA 50219 (manternachk1@central.edu). Research supported by ISU Holl funds. ††Natural and Mathematical Science Division, Culver-Stockton College, Canton, MO 63435 (tpeters319@gmail.com).

ALGORITHMIC PROOF OF MaxMult(T) = p(T)

For a given graph G we consider a set S(G) of all symmetric matrices A = [<TEX>$a_{ij}$</TEX>] whose nonzero entries are placed according to the location of the edges of the graph, i.e., for <TEX>$i{\neq}j$</TEX>, <TEX>$a_{ij}{\neq}0$</TEX> if and only if vertex <TEX>$i$</TEX> is adjacent to vertex <TEX>$j$</TEX>. The minimum rank mr(G) of the graph G is defined to be the smallest rank of a matrix in S(G). In general the computation of mr(G) is complicated, and so is that of the maximum multiplicity MaxMult(G) of an eigenvalue of a matrix in S(G) which is equal to <TEX>$n$</TEX> - mr(G) where n is the number of vertices in G. However, for trees T, there is a recursive formula to compute MaxMult(T). In this note we show that this recursive formula for MaxMult(T) also computes the path cover number <TEX>$p$</TEX>(T) of the tree T. This gives an alternative proof of the interesting result, MaxMult(T) = <TEX>$p$</TEX>(T).

Zero forcing number, maximum nullity, and path cover number of subdivided graphs

The zero forcing number, maximum nullity and path cover number of a (simple, undirected) graph are parameters that are important in the study of minimum rank problems. We investigate the effects on these graph parameters when an edge is subdivided to obtain a so-called edge subdivision graph. An open question raised by Barrett et al. is answered in the negative, and we provide additional evidence for an affirmative answer to another open question in that paper [W. Barrett, R. Bowcutt, M. Cutler, S. Gibelyou, and K. Owens. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.]. It is shown that there is an independent relationship between the change in maximum nullity and zero forcing number caused by subdividing an edge once. Bounds on the effect of a single edge subdivision on the path cover number are presented, conditions under which the path cover number is preserved are given, and it is shown that the path cover number and the zero forcing number of a complete subdivision graph need not be equal.

Zero forcing number, path cover number, and maximum nullity of cacti

The zero forcing number of a graph is the minimum size of a zero forcing set. This parameter is useful in the minimum rank/maximum nullity problem, as it gives an upper bound to the maximum nullity. The path cover number of a graph is the minimum size of a path cover. Results for comparing the parameters are presented, with equality of zero forcing number and path cover number shown for all cacti and equality of zero forcing number and maximum nullity for a subset of cacti. (A cactus is a graph where each edge is in at most one cycle.)

A note on minimum rank and maximum nullity of sign patterns

The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maximum nullity of a zero-nonzero pattern, including path cover number and edit distance, are extended to sign patterns, and the SNS number is introduced to usefully generalize the triangle number to sign patterns. It is shown that for tree sign patterns (that need not be combinatorially symmetric), minimum rank is equal to SNS number, and maximum nullity, path cover number and edit distance are equal, providing a method to compute minimum rank for tree sign patterns. The minimum rank of small sign patterns is determined.

Maximum nullity of outerplanar graphs and the path cover number

Let G = ( V , E ) be a graph with V = { 1 , 2 , … , n } . Define S ( G ) as the set of all n × n real-valued symmetric matrices A = [ a ij ] with a ij ≠ 0 , i ≠ j if and only if ij ∈ E . By M ( G ) we denote the largest possible nullity of any matrix A ∈ S ( G ) . The path cover number of a graph G , denoted P ( G ) , is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G . There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T , M ( T ) = P ( T ) . Barioli et al. [2], show that for a unicyclic graph G , M ( G ) = P ( G ) or M ( G ) = P ( G ) - 1 . Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G , M ( G ) ⩽ P ( G ) . Further we show that for any partial 2-path G , M ( G ) = P ( G ) .

On the minimum rank of not necessarily symmetric matrices: A preliminary study

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ≠ j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G = ( X , Y ; E ) be a balanced bipartite graph of order 2 n . The path-cover number p c ( H ) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H . S ⊆ V ( G ) is called a balanced set of G if | S ∩ X | = | S ∩ Y | . In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S , there is a bi-cycle of every length from | S | + 2 p c ( 〈 S 〉 ) up to 2 n through S .

Smith Normal Form and acyclic matrices

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximal multiplicity MaxMult(T) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T) of distinct eigenvalues over the symmetric matrices whose graphs are T). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).