scholarly journals Extremal Problems for Independent Set Enumeration

10.37236/656 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jonathan Cutler ◽  
A. J. Radcliffe
Keyword(s):  
Initial Segment ◽  
Independence Number ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Extremal Problems ◽  
Fixed Number ◽  
Regular Graphs ◽  
Minimum Number ◽  
Rich History

The study of the number of independent sets in a graph has a rich history. Recently, Kahn proved that disjoint unions of $K_{r,r}$'s have the maximum number of independent sets amongst $r$-regular bipartite graphs. Zhao extended this to all $r$-regular graphs. If we instead restrict the class of graphs to those on a fixed number of vertices and edges, then the Kruskal-Katona theorem implies that the graph with the maximum number of independent sets is the lex graph, where edges form an initial segment of the lexicographic ordering. In this paper, we study three related questions. Firstly, we prove that the lex graph has the maximum number of weighted independent sets for any appropriate weighting. Secondly, we solve the problem of maximizing the number of independents sets in graphs with specified independence number or clique number. Finally, for $m\leq n$, we find the graphs with the minimum number of independent sets for graphs with $n$ vertices and $m$ edges.

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

scholarly journals Hypergraph Independent Sets

2012 ◽  
Vol 22 (1) ◽  
pp. 9-20 ◽  
Author(s):  
JONATHAN CUTLER ◽  
A. J. RADCLIFFE
Keyword(s):  
Upper Bound ◽  
Independent Set ◽  
Independent Sets ◽  
Extremal Problems ◽  
Regularity Lemma ◽  
Uniform Hypergraph ◽  
Fixed Size

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices isj-independentif its intersection with any edge has size strictly less thanj. The Kruskal–Katona theorem implies that in anr-uniform hypergraph with a fixed size and order, the hypergraph with the mostr-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number ofj-independent sets in anr-uniform hypergraph.

Strong vb-dominating and vb-independent sets of a graph

2019 ◽  
Vol 12 (01) ◽  
pp. 2050002 ◽  
Author(s):  
Sayinath Udupa ◽  
R. S. Bhat
Keyword(s):  
Lower Bounds ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Number Formula

Let [Formula: see text] be a graph. A vertex [Formula: see text] strongly (weakly) b-dominates block [Formula: see text] if [Formula: see text] ([Formula: see text]) for every vertex [Formula: see text] in the block [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vb-dominating set (SVBD-set) (WVBD-set) if every block in [Formula: see text] is strongly (weakly) b-dominated by some vertex in [Formula: see text]. The strong (weak) vb-domination number [Formula: see text] ([Formula: see text]) is the order of a minimum SVBD (WVBD) set of [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vertex block independent set (SVBI-set (WVBI-set)) if [Formula: see text] is a vertex block independent set and for every vertex [Formula: see text] and every block [Formula: see text] incident on [Formula: see text], there exists a vertex [Formula: see text] in the block [Formula: see text] such that [Formula: see text] ([Formula: see text]). The strong (weak) vb-independence number [Formula: see text] ([Formula: see text]) is the cardinality of a maximum strong (weak) vertex block independent set (SVBI-set) (WVBI-set) of [Formula: see text]. In this paper, we investigate some relationships between these four parameters. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of the bounds.

scholarly journals On Some Conjectures Concerning Critical Independent Sets of a Graph

10.37236/5580 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Taylor Short
Keyword(s):  
Lower Bound ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Independent Sets ◽  
Simple Graph ◽  
Vertex Set ◽  
The Difference

Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and $\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of Konig-Egervary graphs involving $\text{nucleus}(G)$ and $\text{diadem}(G)$. We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.

scholarly journals Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

2009 ◽  
Vol 19 (1) ◽  
pp. 61-85 ◽  
Author(s):  
DAVID GAMARNIK ◽  
DAVID A. GOLDBERG
Keyword(s):  
Greedy Algorithm ◽  
Greedy Algorithms ◽  
Independent Set ◽  
Independent Sets ◽  
Simple Expression ◽  
Regular Graphs ◽  
Bounded Degree ◽  
Constant Degree ◽  
Image Position ◽  
The Greedy Algorithm

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant-degree regular graphs. We show that forr-regular graphs withnnodes and girth at leastg, the algorithm finds an independent set of expected cardinalitywheref(r) is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degreerand girthgare large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm inarbitrarybounded-degree graphs is concentrated around the mean. Finally, we analyse the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

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