vertex set

Top PDF vertex set:

Faster deterministic feedback vertex set

Faster deterministic feedback vertex set

In our developments, we closely follow the approach of the previously fastest algorithm due to Cao et al. [6]. That is, we first employ the iterative compression technique [24] in a standard manner to reduce the problem to the disjoint compression variant ( Disjoint-FVS ), where the vertex set is split into two parts, both inducing forests, and we are allowed to delete vertices only from the second part. Then we develop a set of reduction and branching rules to cope with this structuralized instance. We rely on the core observation of Cao et al. that the problem becomes polynomial-time solvable once the maximum degree of the deletable vertices drops to 3.
Show more

7 Read more

Simple Proof of Hardness of Feedback Vertex Set

Simple Proof of Hardness of Feedback Vertex Set

Abstract: The Feedback Vertex Set problem (FVS), where the goal is to find a small subset of vertices that intersects every cycle in an input directed graph, is among the fundamental problems whose approximability is not well understood. One can efficiently find an O(logn)- e factor approximation, and efficient constant-factor approximation is ruled out under the Unique Games Conjecture (UGC). We give a simpler proof that Feedback Vertex Set is hard to approximate within any constant factor, assuming UGC.

11 Read more

Linear time parameterized algorithms for subset feedback vertex set

Linear time parameterized algorithms for subset feedback vertex set

It is possible to decompose any graph “in a tree-like fashion” into triconnected parts. While the idea of a triconnected part is similar in spirit to that of a biconnected component, a tri- connected part in our context is not necessarily a subgraph of the input graph and is more conveniently stated using tree decompositions. The concepts of torsos and adhesions are needed to state this decomposition theorem. For a graph G and vertex set M ⊆ V (G), the graph torso(G, M ) has vertex set M . Two vertices u and v have an edge between each other in torso(G, M) if uv ∈ E(G[M]), or there is a path from u to v in G with all internal vertices in V (G) \ M . For a tree decomposition (F, χ) of G and edge ab ∈ E(F ) the set χ(a) ∩ χ(b) is called an adhesion of the tree decomposition (F, χ).
Show more

38 Read more

An Algorithm for the Feedback Vertex Set Problem on a Normal Helly Circular Arc Graph

An Algorithm for the Feedback Vertex Set Problem on a Normal Helly Circular Arc Graph

Let G = ( V E , ) be a simple graph, where V is the set of vertices and E is the set of edges of G, with V = n and E = m . Suppose that V ′ is a nonempty subset of V. The subgraph of G whose vertex set is V ′ and whose edge set is the set of those edges of G that have both vertices in V ′ is called the induced subgraph on V ′ and is denoted by G V [ ] ′ [10]. A cycle with no repeated vertices is a simple cycle. In this paper, the term “cycle” denotes “simple cycle”. A feedback vertex set (FVS) consists of a subset F ⊆ V such that each cycle in G contains at least one vertex in F. In other words, a subset F ⊆ V is an FVS of G if the subgraph induced by G V [ − F ] is acyclic. The FVS problem is to find an FVS of minimum cardinality (MFVS) in G. The FVS problem has applications in several areas such as deadlock prevention in operating systems [11], combinatorial circuit design [12], VLSI circuits [13], and information security [14].
Show more

9 Read more

On group feedback Vertex Set parameterized by the size of the cutset

On group feedback Vertex Set parameterized by the size of the cutset

A comparison with the previous algorithm for Subset Feedback Ver- tex Set [10] is in place; note that the bound on the running time of our algorithm matches the one for Subset Feedback Vertex Set [10], while our algorithm works in a much more general framework. In our opinion, the group-labeled setting is a much more convenient and insightful way of look- ing at graph-separation problems as compared to the definition of Subset Feedback Vertex Set . To support this claim, let us briefly analyse how the algorithm of Theorem 1 solves an ESFVS instance (G, S, k), via the re- duction of Lemma 3. Every edge of S translates to a different Z 2 coordinate
Show more

15 Read more

Minimum Weighted Feedback Vertex Set on Diamonds

Minimum Weighted Feedback Vertex Set on Diamonds

Given a vertex weighted graph G, a minimum Weighted Feedback Vertex Set (MWFVS) is a subset F ⊆ V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The MWFVS on general graph is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve MWFVS on it. We will discuss, moreover, how this result could be used to effectively improve the approximated solution of any known heuristic to solve MWFVS on a general graph.

5 Read more

Efficient Pruning Techniques by Vertex set Similarity and Graph Topology

Efficient Pruning Techniques by Vertex set Similarity and Graph Topology

Accurate subgraph coordinating question requires that all the vertices' and edges are coordinated precisely. The Ullman's subgraph isomorphism technique calculation don't use any file structure, in this way they are typically unreasonable for huge charts. Tree Pi files diagram databases utilizing incessant subtrees as indexing structures. GADDI is a structure separation based subgraph coordinating calculation in a vast diagram. Chao et AL. examined the S Path calculation, which uses most limited ways around the vertex as fundamental record units. Cheng et AL proposed another two-stage R-join calculation to productively discover coordinating chart designs from an expansive diagram. Zou et al. proposed a distance based multi-way join calculation for noting design match questions over an expansive diagram. Shang et al. proposed QuickSI calculation for subgraph isomorphism improved by picking an inquiry request in view of some components of diagrams. SING is a novel indexing framework for subgraph isomorphism in a vast scale chart. GraphQL is a question dialect for chart databases which underpins diagrams as the essential unit of data. Sun et al. used chart investigation and parallel registering to prepare subgraph coordinating inquiry on a billion hub diagram. As of late, a proficient and hearty subgraph isomorphism calculation TurboISO was proposed. RINQ are chart arrangement calculations for organic systems, which can be utilized to take care of isomorphism issues. Be that as it may, an inquiry chart is much littler than the information diagram in subgraph isomorphism issues, while the two charts as a rule have comparable size in chart arrangement issues. To take care of subgraph isomorphism issues, diagram arrangement calculations present extra cost as they ought to first discover competitor subgraphs of comparative size from the extensive information chart. Furthermore, existing accurate subgraph coordinating and chart arrangement calculations don't consider weighted set similitude on vertices, which will bring about high post processing expense of set closeness calculation.
Show more

6 Read more

A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

However, from an implementation point of view, this choice would require both the storage of a large quan- tity of data and the use of complex data structures to manage sets. Now, we describe a more efficient strat- egy to build these optimum sets. In order to do that, we associate with each vertex u of T r a boolean variable

7 Read more

Vol 9, No 4 (2018)

Vol 9, No 4 (2018)

By a graph G = (V, E), we mean a finite undirected connected graph without loops or multiple edges. The order and size of G are denoted by p and q respectively. The distance d(u, v) between two vertices u and v in a connected graph G is the length of a shortest u −v path in G. An u−v path of length d(u, v) is called an u −v geodesic. It is known that the distance is a metric on the vertex set of G. For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. The minimum eccentricity among the vertices of G is the radius, radG and the maximum eccentricity is its diameter, diamG of G. For basic graph theoretic terminology, we refer to Harary [1]. For a nonempty set W of vertices in a connected graph G, the Steiner distance d(W) of W is the minimum size of a connected subgraph of G containing W. Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to W or a Steiner W - tree. It is to be noted that d(W) = d(u, v), when W = {u, v}. If v is an end vertex of a Steiner W-tree, then v ∈ W. Also if < W > is connected, then any Steiner W-tree contains the elements of W only. The set of all vertices of G that lie on some Steiner W-tree is denoted by S(W). If S(W) = V, then W is called a Steiner set for G. A Steiner set of minimum cardinality is a minimum Steiner set or simply a s-set of G and this cardinality is the Steiner number s(G) of G. If W is a Steiner set of G and v a cut vertex of G, then v lies in every Steiner W-tree of G and so W ∪ {v} is also a Steiner set of G. The Steiner number of a graph was introduced in [2] and further studied in [3, 4, 6, 7]. Let x be a vertex of a connected graph G and W ⊂ V (G) such that x ∉ W. Then W is called an x-edge Steiner set of G if every vertex of G lies on some Steiner W ∪ {x} - tree of G. The minimum cardinality of an x-edge Steiner set of G is defined as x-edge Steiner number of G and denoted by s 1x (G). Any x-edge Steiner set of cardinality s 1x (G) is called an s 1x -set of
Show more

5 Read more

Some Properties of Glue Graph

Some Properties of Glue Graph

In general, u is called an eccentric point, if it is an eccentric point of some vertex, otherwise noneccentric. For any graph G, the equi-eccentric point set graph is a graph with vertex set v(G) and two vertices are adjacent if and only if they correspond to two vertices of G with equal eccentricities [5]. The vertex v is a central vertex if e(v)=r(G) and is denoted by ξ r. Let {ξ r } be the set of vertices having minimum eccentricity. A graph

6 Read more

DOMINATION AND EDGE DOMINATION IN TREES

DOMINATION AND EDGE DOMINATION IN TREES

In domination theory, comparison is made between domination parameters defined on vertex set or domination parameters defined on edge set. There are only a few studies on comparison between domination parameter defined on vertex set with a domination parameter defined on edge set, see [4, 8]. Here, a domination parameter defined on edge set, edge domination, is compared with a domination parameter defined on vertex set, vertex domination and we characterize trees with domination number equal to twice edge domination number.

6 Read more

Relatively prime dominating polynomial in graphs

Relatively prime dominating polynomial in graphs

The only minimal relatively prime dominating set of size s+ 1 is obtained by selecting the vertex set D and the vertex v. Therefore, d r pd (G, s + 1) = 1. A relatively prime dominating set of size s +2 is obtained by selecting the vertex set D, a ver- tex from C and the vertex v. This can be done in

8 Read more

A BIPARTITE GRAPH ASSOCIATED WITH IRREDUCIBLE ELEMENTS AND GROUP OF UNITS IN $\mathbb{Z}_n$

A BIPARTITE GRAPH ASSOCIATED WITH IRREDUCIBLE ELEMENTS AND GROUP OF UNITS IN $\mathbb{Z}_n$

and group of units in a ring of integers modulo n. We construct a bipartite graph in which we define a vertex-set as the union of the set of irreducible elements and group of units and an edge-set as the set of pairs between irreducible elements and their unit factors. Our interest is to study and establish relationships between the set of irreducible elements and the group of units by using the properties of the graph we are going to construct. Since we are going to work with elements in the ring of integers modulo n, in Section 2, we give an overview of this ring and we also give some results, available in [7], on how to determine the set of irreducible elements and its cardinality. We also present some results, which are crucial to our work, on group action and graph theory.
Show more

22 Read more

Which fullerenes are stable?

Which fullerenes are stable?

The dual of a fullerene is the plane graph obtained by substituting the roles of vertices and faces: the vertex set of the dual graph is the set of faces of the original graph and two vertices in the dual graph are adjacent if and only if the two faces share an edge in the original graph. The dual of a fullerene with n vertices is a plane graph where every face is a triangle. This operation is called triangulation. The triangulation on a fullerene provides one which contains 12 vertices of degree 5 and n/2 - 10 vertices of degree 6.

7 Read more

Eternal Independent Sets in Graphs

Eternal Independent Sets in Graphs

Let G = (V, E) denote a finite, undirected graph with vertex set V and edge set E. The problem of protecting a graph with mobile guards has been studied in a number of recent papers. We shall begin with a review of some of these models before introducing the eternal independent set problem, which can be viewed in the same light.

12 Read more

THE STRONG (WEAK) VV-DOMINATING SET OF A GRAPH

THE STRONG (WEAK) VV-DOMINATING SET OF A GRAPH

For notations and terminologies refer (Harary, 1969; West, 1996). Let G be a graph of order p and size q. The set of vertices is said to be an independent set if no two vertices in the set are adjacent. Then the maximum order of the independent set is called independence number 𝛽 0 (𝐺). If a vertex v is incident with an edge x then we say that v and x cover each other. The minimum number of vertices required to cover all the edges of G is called vertex covering number 𝛼 0 (𝐺). These two parameters are related by 𝛼 0 (𝐺) + 𝛽 0 (𝐺) = 𝑝 which is now referred as classical Gallai's Theorem (Gallai, 1959). Two vertices are said to dominate each other if they are adjacent. A set of vertices is a dominating set
Show more

10 Read more

Path Induced Geodesic Graphs

Path Induced Geodesic Graphs

Let 𝐺 be a connected graph with at least two vertices. A connected geodetic set 𝑆 ⊆ 𝑉(𝐺) is said to be a path induced geodetic (pig) set of 𝐺 if < 𝑆 > contais a path 𝑃, where 𝑉(𝑃) = 𝑆. The minimum cardinality of a path induced geodetic set of 𝐺 is called a path induced geodetic number of𝐺 and is denoted by 𝑝𝑖𝑔𝑛(𝐺). Some properties satisfied by this concept are studied. It is prove that 𝑝𝑖𝑔𝑛(𝐺) ≥ 1 + 𝑑. It is shown that for any positive integers 2 ≤ 𝑑 < 𝑝, there exists a path induce geodesic graph 𝐺 such that 𝑝𝑖𝑔𝑛(𝐺) = 1 + 𝑑, where 𝑑 is the diameter of 𝐺 and 𝑝 is the order of 𝐺.In this paper we investigate how the path induced geodetic number is affected by adding a pendant edge to 𝐺. It is proved that if 𝐺′ is a graph obtained from 𝐺 by adding a pendant edge, then 𝑝𝑖𝑔𝑛(𝐺 ′ ) ≥ 𝑝𝑖𝑔𝑛(𝐺) + 1.
Show more

8 Read more

A New General Parser for Extensible Languages

A New General Parser for Extensible Languages

The major problem with the multipass method is that for each vertex iteration in a pass, rules are being matched against paths that contain vertices that have not been examined in the current pass. This means matching has not been performed between the ruleset and the majority of the subpaths of the path currently being examined (i.e. all subpaths not containing the left-most vertex).

7 Read more

Graph Theory

Graph Theory

Hamiltonian graph if it admits a Hamiltonian cycle is called Hamiltonian Graph. Hamiltonian graph is also called as Hamilton graph. Hamiltonian Circuit visits each vertex exactly once except for the first vertex, which is also last. A path passing through all the vertices of a graph is called a Hamiltonian path and a graph containing a Hamiltonian path is said to be traceable. Hamiltonian Path does not include any self loop or parallel edge.

14 Read more

Download PDF

Download PDF

Proof. We use induction by using 𝑘 operations to obtain the tree 𝑇. If 𝑇 = 𝑇 = 𝑃 , then 𝛾 (𝑃 ) = 6 3 = 2 ⁄ . Now let 𝑘 is a positive integer. It is assumed that the result is true for every 𝑇 = 𝑇 which is an element of ℱ obtained by 𝑘 − 1 operations. So 𝑛 = 𝑛 − 6. Let 𝑥 a leaf of 𝑇 = 𝑇 which is a path 𝑃 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 is attached by joining one of its leaves to it. Let 𝐷 is 𝛾 (𝑇 )-set. It is easy to see that 𝐷 ∪ {𝑣 , 𝑣 } is a TVEDS of 𝑇. Thus,

7 Read more

Show all 10000 documents...