scholarly journals Excluding Hooks and their Complements

10.37236/6397 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Krzysztof Choromanski ◽  
Dvir Falik ◽  
Anita Liebenau ◽  
Viresh Patel ◽  
Marcin Pilipczuk
Keyword(s):  
Undirected Graph ◽  
Independent Set ◽  
Infinite Family ◽  
Induced Subgraph ◽  
Polynomial Size

The long-standing Erdős-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\epsilon(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or an independent set of size at least $n^{\epsilon(H)}$. A natural weakening of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both $H$ and its complement $H^\mathrm{c}$. These conjectures have been shown to hold for only a handful of graphs: it is not even known if they hold for all graphs on $5$ vertices.In a recent breakthrough, the symmetrized version of the Erdős-Hajnal conjecture was shown to hold for all paths. The goal of this paper is to show that the symmetrized conjecture holds for all trees on $6$ (or fewer) vertices. In fact this is a consequence of showing that the symmetrized conjecture holds for any path with a pendant edge at its third vertex; thus we also give a new infinite family of graphs for which the symmetrized conjecture holds.

scholarly journals Vertex connectivity of the power graph of a finite cyclic group II

2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Vertex Connectivity ◽  
Induced Subgraph ◽  
Power Graph ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
Positive Integers ◽  
Appl Math

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

scholarly journals A New Upper Bound Based on Vertex Partitioning for the Maximum K-plex Problem

2021 ◽  
Author(s):  
Hua Jiang ◽  
Dongming Zhu ◽  
Zhichao Xie ◽  
Shaowen Yao ◽  
Zhang-Hua Fu
Keyword(s):  
Branch And Bound ◽  
Upper Bound ◽  
Undirected Graph ◽  
State Of The Art ◽  
Search Space ◽  
Exact Algorithms ◽  
Induced Subgraph ◽  
Massive Graphs ◽  
Vertex Set ◽  
Number Of Branches

Given an undirected graph, the Maximum k-plex Problem (MKP) is to find a largest induced subgraph in which each vertex has at most k−1 non-adjacent vertices. The problem arises in social network analysis and has found applications in many important areas employing graph-based data mining. Existing exact algorithms usually implement a branch-and-bound approach that requires a tight upper bound to reduce the search space. In this paper, we propose a new upper bound for MKP, which is a partitioning of the candidate vertex set with respect to the constructing solution. We implement a new branch-and-bound algorithm that employs the upper bound to reduce the number of branches. Experimental results show that the upper bound is very effective in reducing the search space. The new algorithm outperforms the state-of-the-art algorithms significantly on real-world massive graphs, DIMACS graphs and random graphs.

scholarly journals A Parameterized Complexity View on Collapsing k-Cores

2021 ◽  
Author(s):  
Junjie Luo ◽  
Hendrik Molter ◽  
Ondřej Suchý
Keyword(s):  
Parameterized Complexity ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Structural Parameters ◽  
Vertex Cover ◽  
Polynomial Kernel ◽  
Input Graph ◽  
Induced Subgraph ◽  
Vertex Deletion ◽  
Drop Outs

AbstractWe study the -hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. (2017) and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be -hard for all r ≥ 0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r. We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k ≤ 2 and k ≥ 3. For the latter case it is known that for all x ≥ 0 Collapsed k-Core is -hard when parameterized by b. For k ≤ 2 we show that Collapsed k-Core is -hard when parameterized by b and in when parameterized by (b + x). Furthermore, we outline that Collapsed k-Core is in when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

scholarly journals Induced Paths in Twin-Free Graphs

10.37236/892 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
David Auger
Keyword(s):  
Undirected Graph ◽  
Induced Subgraph ◽  
Identifying Codes ◽  
Chordless Path

Let $G=(V,E)$ be a simple, undirected graph. Given an integer $r \geq 1$, we say that $G$ is $r$-twin-free (or $r$-identifiable) if the balls $B(v,r)$ for $v \in V$ are all different, where $B(v,r)$ denotes the set of all vertices which can be linked to $v$ by a path with at most $r$ edges. These graphs are precisely the ones which admit $r$-identifying codes. We show that if a graph $G$ is $r$-twin-free, then it contains a path on $2r+1$ vertices as an induced subgraph, i.e. a chordless path.

scholarly journals Algorithmic Aspects of Some Variants of Domination in Graphs

2020 ◽  
Vol 28 (3) ◽  
pp. 153-170
Author(s):  
J. Pavan Kumar ◽  
P.Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Bipartite Graphs ◽  
Minimum Size ◽  
Induced Subgraph ◽  
Split Graphs ◽  
Np Complete ◽  
Secure Domination

AbstractA set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γis(G). Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we show that the MInSDS problem is APX-hard for graphs with maximum degree 5.

Automorphism group and fixing number of the orthogonality graph based on rank one upper triangular matrices

2020 ◽  
pp. 2250023
Author(s):  
Shikun Ou ◽  
Yanqi Fan ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Canonical Form ◽  
Undirected Graph ◽  
Induced Subgraph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Rank One ◽  
Vertex Set

The orthogonality graph [Formula: see text] of a ring [Formula: see text] is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of [Formula: see text], in which for two vertices [Formula: see text] and [Formula: see text] (needless distinct), [Formula: see text] is an edge if and only if [Formula: see text]. Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of [Formula: see text], the induced subgraph of [Formula: see text] with vertex set [Formula: see text].

scholarly journals Induced Subgraphs With Many Distinct Degrees

2017 ◽  
Vol 27 (1) ◽  
pp. 110-123 ◽  
Author(s):  
BHARGAV NARAYANAN ◽  
ISTVÁN TOMON
Keyword(s):  
Analogous Result ◽  
Independent Set ◽  
Independent Sets ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Vertex Graph ◽  
Fixed Constant

Let hom(G) denote the size of the largest clique or independent set of a graphG. In 2007, Bukh and Sudakov proved that everyn-vertex graphGwith hom(G) =O(logn) contains an induced subgraph with Ω(n1/2) distinct degrees, and raised the question of deciding whether an analogous result holds for everyn-vertex graphGwith hom(G) =O(nϵ), whereϵ> 0 is a fixed constant. Here, we answer their question in the affirmative and show that every graphGonnvertices contains an induced subgraph with Ω((n/hom(G))1/2) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixedk∈ ℕ, that if ann-vertex graphGcontains no induced subgraph withkdistinct degrees, then hom(G)⩾n/(k− 1) −o(n); this bound is essentially best possible.

scholarly journals On r-Noncommuting Graph of Finite Rings

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 233
Author(s):  
Rajat Kanti Nath ◽  
Monalisha Sharma ◽  
Parama Dutta ◽  
Yilun Shang
Keyword(s):  
Undirected Graph ◽  
Clique Number ◽  
Star Graph ◽  
Finite Rings ◽  
Induced Subgraph ◽  
Noncommutative Rings ◽  
Central Elements ◽  
Vertex Set ◽  
Vertex Degrees ◽  
Noncommuting Graph

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

scholarly journals Maximum difference about the size of optimal identifying codes in graphs differing by one vertex

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mikko Pelto
Keyword(s):  
Upper Bound ◽  
Undirected Graph ◽  
Minimum Size ◽  
Maximum Difference ◽  
Free Graph ◽  
Induced Subgraph ◽  
Identifying Code ◽  
Vertex Set ◽  
International Audience ◽  
Codes In Graphs

Graph Theory International audience Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.

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