Path Partition Research Articles

On a tree-partition problem

If T=(V,E) is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees T1, T2, so that V(T1)∩V(T2)={v}, V(T1)∪(T2)=V(T), E(T1)∩E(T2)=∅, E(T1)∪E(T2)=E(T), T[V(T1)\{v}]⊆E(T1) and T[V(T2)\{v}]⊆E(T2). We call such a partition a Tv−partition of T. We study the following em: Given a graph G belonging to –T. Is it true that for any Tv-partition T1, T2 of T there exists a partition {V1,V2} of the vertices of G such that G[V1]∈−T1 and G[V2]∈−T2? This problem provides a natural generalization of Δ-partition problem studied by L. Lovász ([L. Lovász, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihók ([P. Mihók, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Góra, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture.

An efficient algorithm for the minimum L2 path partition

An efficient algorithm for the minimum L2 path partition

Proof of Berge’s strong path partition conjecture for k=2

Berge’s strong path partition conjecture from 1982 generalizes and extends Dilworth’s theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture is known to be true for all digraphs only for k = 1 (by the Gallai–Milgram theorem) and for k ≥ λ (where λ is the cardinality of the longest path in the graph). The attempts made, so far, to prove the conjecture for other values of k have yielded proofs for acyclic digraphs, but not for general digraphs. In this paper, we prove the conjecture for k = 2 for all digraphs. The proof is constructive and it extends the proof for k = 1 .

The path partition problem and related problems in bipartite graphs

We prove that it is NP-complete to decide whether a bipartite graph of maximum degree three on nk vertices can be partitioned into n paths of length k. Finally, we propose some approximation and inapproximation results for several related problems.

The Path Partition Conjecture is true for claw-free graphs

The detour order of a graph G , denoted by τ ( G ) , is the order of a longest path in G . The Path Partition Conjecture (PPC) is the following: If G is any graph and ( a , b ) any pair of positive integers such that τ ( G ) = a + b , then the vertex set of G has a partition ( A , B ) such that τ ( 〈 A 〉 ) ⩽ a and τ ( 〈 B 〉 ) ⩽ b . We prove that this conjecture is true for the class of claw-free graphs. We also show that to prove that the PPC is true, it is sufficient to consider the class of 2-connected graphs.

Minmax relations for cyclically ordered digraphs

We prove a range of minmax theorems about cycle packing and covering in digraphs whose vertices are cyclically ordered, a notion promoted by Bessy and Thomassé in their beautiful proof of the following conjecture of Gallai: the vertices of a strongly connected digraph can be covered by at most as many cycles as the stability number. The results presented here provide relations between cycle packing and covering and various objects in graphs such as stable sets, their unions, or feedback vertex- and arc-sets. They contain the results of Bessy and Thomassé with simple algorithmic proofs, including polynomial algorithms for weighted variants, classical results on posets extending Greene and Kleitman's theorem (that in turn contains Dilworth's theorem), and a common generalization of these. The most general minmax results concern the maximum number of vertices of a k-chromatic subgraph ( k ∈ N ) —as a consequence, this number is greater than or equal to the minimum of | X | + k | C | , running on subsets of vertices X and families of cycles C covering all vertices not in X, in strongly connected digraphs. This is the “circuit cover” version of a conjecture of Linial (like Gallai's conjecture is the circuit cover version of the Gallai–Milgram theorem, meaning that path partitions are replaced by circuit covers under strong connectivity); we also deduce the circuit cover version of a conjecture of Berge on path partitions; these conjectures remain open, but the proven statements also bound the maximum size of a k-chromatic subgraph, contain Gallai's conjecture, and Bessy and Thomassé's theorems. All presented minmax relations are proved using cyclic orders and a unique elementary argument based on network flows—algorithmically only shortest paths and potentials in conservative digraphs—varying the parameters of the network flow model. In this way antiblocking and blocking relations can be established, leading to a general polyhedral phenomenon—a combination of “integer decomposition, integer rounding” and “dual integrality”—that also contains the matroid partition theorem and Dilworth's theorem. We provide a common reason for all these minmax equalities to hold and some possible other ones which satisfy the same abstract properties.

Berge's conjecture on directed path partitions—a survey

Berge's conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth's theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k ⩾ 1 . The conjecture is still open and intriguing for all k > 1 . 1 1 Only recently it was proved Berger and Ben-Arroyo Hartman [56] for k = 2 (added in proof). In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians.

Longest path partitions in generalizations of tournaments

We consider the so-called Path Partition Conjecture for digraphs which states that for every digraph, D, and every choice of positive integers, λ 1 , λ 2 , such that λ 1 + λ 2 equals the order of a longest directed path in D, there exists a partition of D into two digraphs, D 1 and D 2 , such that the order of a longest path in D i is at most λ i , for i = 1 , 2 . We prove that certain classes of digraphs, which are generalizations of tournaments, satisfy the Path Partition Conjecture and that some of the classes even satisfy the conjecture with equality.

An Asymptotic Result for the Path Partition Conjecture

The detour order of a graph $G$, denoted by $\tau \left( G\right) ,$ is the order of a longest path in $G.$ A partition of the vertex set of $G$ into two sets, $A$ and $B,$ such that $\tau (\left\langle A\right\rangle )\leq a$ and $\tau (\left\langle B\right\rangle )\leq b$ is called an $(a,b)$-partition of $G$. If $G$ has an $(a,b)$-partition for every pair $(a,b)$ of positive integers such that $a+b=\tau (G),$ then we say that $G$ is $\tau $-partitionable. The Path Partition Conjecture (PPC), which was discussed by Lovász and Mihók in 1981 in Szeged, is that every graph is $\tau $-partitionable. It is known that a graph $G$ of order $n$ and detour order $\tau =n-p$ is $\tau $-partitionable if $p=0,1.$ We show that this is also true for $p=2,3,$ and for all $p\geq 4$ provided that $n\geq p(10p-3).$

The directed path partition conjecture

The directed path partition conjecture

A partition index for XML and semi-structured data*1, *2

XML and other semi-structured data can be represented by a graph model. The paths in a data graph are used as a basic constructor of a query. Especially, by using patterns on paths, a user can formulate more expressive queries. Patterns in a path enlarge the search space of a data graph and current research for indexing semi-structured data focuses on reducing the search space. However, the existing indexes cannot reduce the search space when a data graph has some references. In this paper, we introduce a partitioning technique for all paths in a data graph and an index graph which can effectively find appropriate path partitions for a path query with patterns.

Path partitions and [formula omitted]-free sets

The detour order τ ( G ) of a graph G is the order of a longest path of G. If S is a subset of V ( G ) such that the graph induced by S has detour order at most n, then S is called a P n + 1 - free set in G. The Path Partition Conjecture (PPC) can be stated as follows: For any graph G and any positive integer n < τ ( G ) , there exists a P n + 1 -free set H in G such that τ ( G - H ) ⩽ τ ( G ) - n . We prove that if G is any graph and M is any maximal P n + 1 -free set in G, then τ ( G - M ) ⩽ τ ( G ) - 2 3 ( n + 1 ) . We also prove that if G has no cycle of order less than n or greater than τ ( G ) - n + 2 , then τ ( G - M ) ⩽ τ ( G ) - n for every maximal P n + 1 -free subset M of G . As a corollary of the latter result we prove that the PPC is true for the class of connected, weakly pancyclic graphs.

Path partition for graphs with special blocks

The path-partition problem is to find a minimum number of vertex-disjoint paths that cover all vertices of a given graph. This paper studies the path-partition problem from an algorithmic point of view. As the Hamiltonian path problem is NP-complete for many classes of graphs, so is the path-partition problem. The main result of this paper is to present a linear-time algorithm for the path-partition problem in graphs whose blocks are complete graphs, cycles or complete bipartite graphs.

A partition index for XML and semi-structured data

XML and other semi-structured data can be represented by a graph model. The paths in a data graph are used as a basic constructor of a query. Especially, by using patterns on paths, a user can formulate more expressive queries. Patterns in a path enlarge the search space of a data graph and current research for indexing semi-structured data focuses on reducing the search space. However, the existing indexes cannot reduce the search space when a data graph has some references. In this paper, we introduce a partitioning technique for all paths in a data graph and an index graph which can effectively find appropriate path partitions for a path query with patterns.

Graphs with not all possible path-kernels

The Path Partition Conjecture states that the vertices of a graph G with longest path of length c may be partitioned into two parts X and Y such that the longest path in the subgraph of G induced by X has length at most a and the longest path in the subgraph of G induced by Y has length at most b, where a+ b= c. Moreover, for each pair a, b such that a+ b= c there is a partition with this property. A stronger conjecture by Broere, Hajnal and Mihók, raised as a problem by Mihók in 1985, states the following: For every graph G and each integer k, c⩾ k⩾2 there is a partition of V( G) into two parts (K, K ̄ ) such that the subgraph G[ K] of G induced by K has no path on more than k−1 vertices and each vertex in K ̄ is adjacent to an endvertex of a path on k−1 vertices in G[ K]. In this paper we provide a counterexample to this conjecture.

Path partitions and forward-only trellis algorithms

This is a semitutorial paper on trellis-based algorithms. We argue that most decoding/detection algorithms described on trellises can be formulated as path-partitioning algorithms, with proper definitions of mappings from subsets of paths to metrics of subsets. Thereby, the only two operations needed are path-concatenation and path-collection, which play the roles of multiplication and addition, respectively. Furthermore, we show that the trellis structure permits the path-partitioning algorithms to be formulated as forward-only algorithms (with structures resembling the Viterbi (1967) algorithm), thus eliminating the need for backward computations regardless of what task needs to be performed on the trellis. While all of the actual decoding/detection algorithms presented here are rederivations of variations of previously known methods, we believe that the exposition of the algorithms in a unified manner as forward-only path-partitioning algorithms is the most intuitive manner in which to generalize the Viterbi algorithm. We also believe that this approach may, in fact, influence the practical implementation of the algorithms as well as influence the construction of other forward-only algorithms (e.g., byte-wise forward-only detection algorithms).

Path Partitions and Pn-free Sets

Abstract The detour order τ(G) of a graph G is the order of a longest path of G. A partition (A, B) of V is called an (a, b)-partition of G if τ(G[A]) ≤ a and τ(G[B]) ≤ b. The Path Partition Conjecture is the following: For any graph G, with detour order τ(G) = a + b, there exists an (a, b)-partition of G. We introduce and examine a conjecture which is possibly stronger: If M is a maximum Pn+1-free set of vertices of G, with n

Optimal path cover problem on block graphs

Let G = ( V, E) be a block graph. First we show that an algorithm for finding the path partition number p( G) by J.H. Yan and G.J. Chang gives wrong answers to some block graphs. Then we present an efficient algorithm for finding a minimum path partition of G (not just the path partition number p( G)). The complexity of this algorithm is O( m), where m = ¦E¦.

On mixed Ramsey numbers

For a graph theoretic parameter f, an integer m and a graph H, the mixed Ramsey number v(f; m; H) is defined as the least positive integer p such that if G is any graph of order p, then either f(G)⩾m or Ḡ contains a subgraph isomorphic to H. Let ϱ denote vertex linear arboricity and let H be any connected graph of order n. In this note we show that v(ϱ; m; H) = 1 + (n + np(H) − 2)(m − 1), where np(H) is the path partition number of H.

THE PATH-PARTITION PROBLEM IN BIPARTITE DISTANCE-HEREDITARY GRAPHS

A path partition of a graph is a collection of vertex-disjoint paths that cover all vertices of the graph. The path-partition problem is to nd a path partition of minimum size. This paper gives a linear-time algorithm for the path-partition problem in bipartite distance-hereditary graphs.