Union Of Paths Research Articles

On the homotopy type of multipath complexes

AbstractA multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph is the simplicial complex whose faces are the multipaths of . We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.

Partitioning planar graphs without 4-cycles and 5-cycles into two forests with a specific condition

Let A be the family of planar graphs without 4-cycles and 5-cycles. In 2013, Hill et al. proved that every graph G∈A has a partition dividing V(G) into three sets, where two of them are independent, and the other induces a graph with a maximum degree at most 3.In 2021 Cho, Choi, and Park conjectured that every graph G∈A has a partition dividing V(G) into two sets, where one set induces a forest, and the other induces a forest with a maximum degree at most 2.In this paper, we show that every graph G∈A has a partition dividing V(G) into two sets, where one set induces a forest, and the other induces a disjoint union of paths and subdivisions of K1,3. The result improves the aforementioned result by Hill et al. and yields progress toward the conjecture of Cho, Choi, and Park.

Complete bipartite graphs without small rainbow stars

The k-edge-colored bipartite Gallai–Ramsey number bgrk(G:H) is defined as the minimum integer n such that n2≥k and for every N≥n, every edge-coloring (using all k colors) of complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first study the structural theorem on the complete bipartite graph Kn,n with no rainbow copy of K1,3. Next, we utilize the results to prove the exact values of bgrk(P4:H), bgrk(P5:H), bgrk(K1,3:H), where H is a various union of cycles and paths and stars.

Preconditioning Sparse Matrices with Alternating and Multiplicative Operator Splittings

We present an algebraic framework for operator splitting preconditioners for general sparse matrices. The framework leads to four different approaches: two with alternating splittings and two with a multiplicative ansatz. The ansatz generalizes ADI and ILU methods to multiple factors and to a more general factor form. The factors may be computed directly from the matrix coefficients or adaptively by incomplete sparse inversions. The special case of tridiagonal splittings is examined in more detail. We decompose the adjacency graph of the sparse matrix into multiple (almost) disjoint linear forests, and each linear forest (a union of disjoint paths) leads to a tridiagonal splitting. We obtain specialized variants of the four general approaches. Parallel implementations for all steps are provided on a GPU. We demonstrate the effectiveness and efficiency of these preconditioners combined with GMRES on various matrices.

Ramsey and Gallai–Ramsey numbers for the union of paths and stars

Given k graphs H 1 , H 2 , … , H k , the k -color Ramsey number R ( H 1 , H 2 , … , H k ) is defined as the minimum integer n such that every k -edge-coloring of K n contains a monochromatic H i , for some i ∈ [ k ] . If H 1 = H 2 = … = H k = H , then we write the number as R k ( H ) . Given two graphs G and H , and a positive integer k , define the Gallai–Ramsey number gr k ( G : H ) as the minimum number of vertices n such that any exact k -edge coloring of K n contains either a rainbow copy of G or a monochromatic copy of H . Much like Ramsey numbers, Gallai–Ramsey numbers have gained a reputation as being difficult to compute in general. In this paper, we determine the exact values or upper and lower bounds of Gallai–Ramsey number gr k ( G : H ) and Ramsey numbers R 2 ( H ) , R 3 ( H ) when H is the union of a path and a star, and G is a 3-star or a 4-path or P 4 + , where P 4 + is the graph consisting of a P 4 with one extra edge incident with an inner vertex.

The Maximum Number of Stars in a Graph Without Linear Forest

For two graphs J and H, the generalized Turán number, denoted by ex(n, J, H), is the maximum number of copies of J in an H-free graph of order n. A linear forest F is the disjoint union of paths. In this paper, we determine $$ex(n,S_r,F)$$ when n is large enough, which generalizes the results on $$ex(n, S_r, P_k)$$ and $$ex(n, (k+1) P_2)$$ . Finally, we prose a problem related to the number of graph copies in an F-free graph under shifting operations.

On Distance Irregular Labeling of Disconnected Graphs

A distance irregular k-labeling of a graph G is a function f : V (G) → {1, 2, . . . , k} such that the weights of all vertices are distinct. The weight of a vertex v, denoted by wt(v), is the sum of labels of all vertices adjacent to v (distance 1 from v), that is, wt(v) = P u∈N(v) f(u). If the graph G admits a distance irregular labeling then G is called a distance irregular graph. The distance irregularity strength of G is the minimum k for which G has a distance irregular k-labeling and is denoted by dis(G). In this paper, we derive a new lower bound of distance irregularity strength for graphs with t pendant vertices. We also determine the distance irregularity strength of some families of disconnected graphs namely disjoint union of paths, suns, helms and friendships.

Linear Forest mP3 Plus a Longer Path Pn Becoming Antimagic

An edge labeling of a graph G is a bijection f from E(G) to a set of |E(G)| integers. For a vertex u in G, the induced vertex sum of u, denoted by f+(u), is defined as f+(u)=∑uv∈E(G)f(uv). Graph G is said to be antimagic if it has an edge labeling g such that g(E(G))={1,2,⋯,|E(G)|} and g+(u)≠g+(v) for any pair u,v∈V(G) with u≠v. A linear forest is a union of disjoint paths of orders greater than one. Let mPk denote a linear forest consisting of m disjoint copies of path Pk. It is known that mP3 is antimagic if and only if m=1. In this study, we add a disjoint path Pn (n≥4) to mP3 and develop a necessary condition and a sufficient condition whereby the new linear forest mP3⋃Pn may be antimagic.

The elastic and directed percolation backbone

We argue that the elastic backbone (EB) (union of shortest paths) on a cylindrical system, studied by Sampaio Filho et al [2018 Phys. Rev. Lett. 120 175701], is in fact the backbone of two-dimensional directed percolation (DP). We simulate the EB on the same system as considered by these authors, and also study the DP backbone directly using an algorithm that allows backbones to be generated in a completely periodic manner. We find that both the EB in the bulk and the DP backbone have a fractal dimension of d b = d B,DP = 1.681 02(15) at the identical critical point p c,DP ≈ 0.705 485 22. We also measure the fractal dimension at the edge of the EB system and for the full DP clusters, and find d e = d DP = 1.840 54(4). We argue that those two fractal dimensions follow from the DP exponents as d B,DP = 2 − 2β/ν ∥ = 1.681 072(12) and d DP = 2 − β/ν ∥ = 1.840 536(6). Our fractal dimensions differ from the value 1.750(3) found by Sampaio Filho et al.

Sum divisor cordial labeling of disjoint union of paths and subdivided star

A bijective map ρ from V(Ω) →{1,2, … |V(Ω)|} is called sum divisor cordial labeling for graph Ω so that for every uυ ∈ E(Ω) edge is fixed the label 1 if 2 divides ρ (u) + ρ(υ) and 0 otherwise, then the difference between number of edges labeled with 1 and the number of edges labeled with 0 by at most 1. A graph is called sum divisor cordial graph if it admits sum divisor cordial labeling. In present article, we investigate the disconnected graph such as the disjoint union of paths and subdivided star are sum divisor cordial graphs.

LazyB: fast and cheap genome assembly

BackgroundAdvances in genome sequencing over the last years have lead to a fundamental paradigm shift in the field. With steadily decreasing sequencing costs, genome projects are no longer limited by the cost of raw sequencing data, but rather by computational problems associated with genome assembly. There is an urgent demand for more efficient and and more accurate methods is particular with regard to the highly complex and often very large genomes of animals and plants. Most recently, “hybrid” methods that integrate short and long read data have been devised to address this need.ResultsLazyB is such a hybrid genome assembler. It has been designed specificially with an emphasis on utilizing low-coverage short and long reads. LazyB starts from a bipartite overlap graph between long reads and restrictively filtered short-read unitigs. This graph is translated into a long-read overlap graph G. Instead of the more conventional approach of removing tips, bubbles, and other local features, LazyB stepwisely extracts subgraphs whose global properties approach a disjoint union of paths. First, a consistently oriented subgraph is extracted, which in a second step is reduced to a directed acyclic graph. In the next step, properties of proper interval graphs are used to extract contigs as maximum weight paths. These path are translated into genomic sequences only in the final step. A prototype implementation of LazyB, entirely written in python, not only yields significantly more accurate assemblies of the yeast and fruit fly genomes compared to state-of-the-art pipelines but also requires much less computational effort.ConclusionsLazyB is new low-cost genome assembler that copes well with large genomes and low coverage. It is based on a novel approach for reducing the overlap graph to a collection of paths, thus opening new avenues for future improvements.AvailabilityThe LazyB prototype is available at https://github.com/TGatter/LazyB.

이동 시간의 불확실성을 고려한 Hyper-path 최적화 알고리즘

We consider a hyper-path optimization problem under travel time uncertainty. In the multimodal networks, the waiting time to take transit should be considered in determining the optimal route to the destination. The hyper-path is a union of simple paths between the origin and destination, which enables the traveler to reduce the total travel time by taking more than one route into account. We extend the hyper-path problem to the case where the travel times are uncertain, so the reliability of the hyper-path should be addressed. We formulate the problem as a nonlinear integer programming problem and develop a heuristic algorithm based on the column generation method. The presented algorithm was verified on the real-life public transportation network with an extensive computational study. A proof-of-concept app for recommending public transportation routes utilizing the proposed algorithm was also developed.

Ramsey numbers of path-matchings, covering designs, and 1-cores

A path-matching of order p is a vertex disjoint union of nontrivial paths spanning p vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers r,p1,…,pr, define RPM(p1,…,pr) to be the smallest integer n such that in any r-coloring of the edges of Kn there exists a path-matching of color i and order at least pi for some i∈[r]. Our main result is that for r≥2 and p1≥…≥pr≥2, if p1≥2r−2, thenRPM(p1,…,pr)=p1−(r−1)+∑i=2r⌈pi3⌉. Perhaps surprisingly, we show that when p1<2r−2, it is possible that RPM(p1,…,pr) is larger than p1−(r−1)+∑i=2r⌈pi3⌉, but in any case we determine the correct value to within a constant (depending on r); i.e.p1−(r−1)+∑i=2r⌈pi3⌉≤RPM(p1,…,pr)≤⌈p1−r3+∑i=2rpi3⌉. As a corollary we get that in every r-coloring of Kn there is a monochromatic path-matching of order at least 3⌊nr+2⌋, which is essentially best possible. We also determine RPM(p1,…,pr) in all cases when the number of colors is at most 4.The proof of the main result uses a minimax theorem for path-matchings derived from a result of Las Vergnas (extending Tutte's 1-factor theorem) to show that the value of RPM(p1,…,pr) depends on the block sizes in covering designs (which can be also formulated in terms of monochromatic 1-cores in colored complete graphs). While block sizes in covering designs have been studied intensively before, they seem to have only been studied in the uniform case (when all block sizes are equal). Then we obtain the result above by giving estimates on the block sizes in covering designs in the arbitrary (non-uniform) case.

RSEMT labeling of union of paths and subdivided star

A reverse edge magic total (REMT) labeling for the graph Γ = (V(Γ), E(Γ)) along with cardinality of p = |V(G)| and q = |E(G)| respectively. A one-one map π: V(Γ) ∪ E(Γ) → {1, 2, … ,|V(Γ)|+|E(Γ)|} having a rule that for every edge, uv ϵ E(Γ), then π (uv) − {π (u) + π (v)} = c, where c is a magic number. A reverse edge magic total (REMT) labeling is called a reverse super edge magic total (RSEMT) labeling if vertex-set receive natural numbers. In present paper, we compute the RSEMT labeling on the forest of paths and subdivided star.

Linear Arboricity of Outer-1-Planar Graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. Zhang et al. (Edge covering pseudo-outerplanar graphs with forests, Discrete Math 312:2788–2799, 2012; MR2945171) proved that the linear arboricity of every outer-1-planar graph with maximum degree $$\Delta $$ is exactly $$\lceil \Delta /2\rceil $$ provided that $$\Delta =3$$ or $$\Delta \geqslant 5$$ and claimed that there are outer-1-planar graphs with maximum degree $$\Delta =4$$ and linear arboricity $$\lceil (\Delta +1)/2\rceil =3$$ . It is shown in this paper that the linear arboricity of every outer-1-planar graph with maximum degree 4 is exactly 2 provided that it admits an outer-1-planar drawing with crossing distance at least 1 and crossing width at least 2, and moreover, none of the above constraints on the crossing distance and crossing width can be removed. Besides, a polynomial-time algorithm for constructing a path-2-coloring (i.e., an edge 2-coloring such that each color class induces a linear forest, a disjoint union of paths) of such an outer-1-planar drawing is given.

Cutting lemma and union lemma for the domination game

Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of a suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are listed.

Path Choosability of Planar Graphs

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

Game domination numbers of a disjoint union of paths and cycles

The domination game is played on a graph G by two players, Dominator and Staller, who alternately chooses a vertex of G in such a way that at least one new vertex is dominated. The game ends when all vertices are dominated. Dominator aims to finish the game in as few moves as possible while Staller aims to finish the game in as many moves as possible. The game domination number γg (G) (respectively γʹg (G)) is the total number of moves both players use in a game which Dominator (respectively Staller) starts and both players use optimal strategies. In this paper we determine the game domination numbers of a disjoint union of paths and cycles.

On Properties of Optimal Paths in First-Passage Percolation

In this paper, we study some properties of optimal paths in the first passage percolation on $\Z^d$ and show the followings: (1) the number of optimal paths has an exponential growth if the distribution has an atom; (2) the means of intersection and union of optimal paths are linear in the distance. For the proofs, we use the configuration--flipping argument introduced in [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993] with suitable adaptions.

An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles

An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles