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
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

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

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
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

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.

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
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

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
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

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.

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

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

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.

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

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

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
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**).

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

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,