ON GRAPHS WITH LIMITED NUMBER OF P4-PARTNERS

1999 ◽  
Vol 10 (01) ◽  
pp. 103-121 ◽  
Author(s):  
FLORIAN ROUSSEL ◽  
IRENA RUSU ◽  
HENRI THUILLIER
Keyword(s):  
Chromatic Number ◽  
Optimization Problems ◽  
Linear Time ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Stability Number

The study of graphs containing few P4's generated an important number of results related to perfection, recognition, optimization problems (see [12], [15], [8]). We define here a new, larger class of graphs and show that the indicated problems may be efficiently solved on this class too (thus generalizing some of the previous results). Namely, we give a linear time recognition algorithm for this class and we note that the optimization problems concerning the clique number, stability number, chromatic number and clique cover number are solvable in linear time.

scholarly journals On P_4-tidy graphs

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
V. Giakoumakis ◽  
F. Roussel ◽  
H. Thuillier
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Hamiltonian Path ◽  
Clique Number ◽  
Stability Number ◽  
Sparse Graphs ◽  
Modular Decomposition ◽  
New Class ◽  
International Audience ◽  
Linear Algorithms

International audience We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].

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 Recognition and Optimization Algorithms for P5-Free Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 304
Author(s):  
Mihai Talmaciu ◽  
Luminiţa Dumitriu ◽  
Ioan Şuşnea ◽  
Victor Lepin ◽  
László Barna Iantovics
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Independent Set ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Feedback Vertex Set ◽  
Free Graph ◽  
Dominating Number ◽  
Vertex Set ◽  
Np Complete

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

scholarly journals Getting new algorithmic results by extending distance-hereditary graphs via split composition

2021 ◽  
Vol 7 ◽  
pp. e627
Author(s):  
Serafino Cicerone ◽  
Gabriele Di Stefano
Keyword(s):  
Chromatic Number ◽  
Domination Number ◽  
Graph Isomorphism ◽  
Clique Number ◽  
Efficient Algorithms ◽  
Combinatorial Problems ◽  
Stability Number ◽  
New Class ◽  
Graph Classes ◽  
Graph Class

In this paper, we consider the graph class denoted as Gen(∗;P3,C3,C5). It contains all graphs that can be generated by the split composition operation using path P3, cycle C3, and any cycle C5 as components. This graph class extends the well-known class of distance-hereditary graphs, which corresponds, according to the adopted generative notation, to Gen(∗;P3,C3). We also use the concept of stretch number for providing a relationship between Gen(∗;P3,C3) and the hierarchy formed by the graph classes DH(k), with k ≥1, where DH(1) also coincides with the class of distance-hereditary graphs. For the addressed graph class, we prove there exist efficient algorithms for several basic combinatorial problems, like recognition, stretch number, stability number, clique number, domination number, chromatic number, and graph isomorphism. We also prove that graphs in the new class have bounded clique-width.

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 Entropy-Penalized Semidefinite Programming

2019 ◽  
Author(s):  
Mikhail Krechetov ◽  
Jakub Marecek ◽  
Yury Maximov ◽  
Martin Takac
Keyword(s):  
Machine Learning ◽  
Time Complexity ◽  
Optimization Problems ◽  
Linear Time ◽  
Broad Class ◽  
Low Rank ◽  
Learning Problems ◽  
Unified Framework ◽  
Gradient Computation ◽  
Machine Learning Applications

Low-rank methods for semi-definite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schatten-norm penalties, which are difficult to implement in practice due to high computational efforts. In this paper, we propose Entropy-Penalized Semi-Definite Programming (EP-SDP), which provides a unified framework for a broad class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit an efficient numerical algorithm, having (almost) linear time complexity of the gradient computation; this makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

scholarly journals On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

10.37236/1805 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Alexandr Kostochka ◽  
Kittikorn Nakprasit
Keyword(s):  
Chromatic Number ◽  
Convex Sets ◽  
Convex Set ◽  
Clique Number ◽  
Finite Family ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Homothetic Copies

Let $G$ be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number $k$ is $(3k-3)$-degenerate. This bound is sharp. As a consequence, we derive that $G$ is $(3k-2)$-colorable. We show also that the chromatic number of every intersection graph $H$ of a family of homothetic copies of a fixed convex set in the plane with clique number $k$ is at most $6k-6$.

scholarly journals Minimum Clique Number, Chromatic Number, and Ramsey Numbers

10.37236/2125 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Chromatic Number ◽  
Asymptotic Properties ◽  
Inverse Function ◽  
Clique Number ◽  
Ramsey Number ◽  
Growth Order ◽  
Ramsey Numbers

Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We investigate the asymptotics of $Q(n,c)$ when $n/c$ is held constant. We show that when $n/c$ is an integer $\alpha$, $Q(n,c)$ has the same growth order as the inverse function of the Ramsey number $R(\alpha+1,t)$ (as a function of $t$). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers hold, then $Q(n,c)$ is in fact asymptotically equivalent to the aforementioned inverse function. We use this fact to deduce that $Q(n,\lceil n/3 \rceil)$ is asymptotically equivalent to the inverse function of $R(4,t)$.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari ◽  
M. Behboodi ◽  
R. Nikandish ◽  
M. J. Nikmehr ◽  
...  
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

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