Balanced Permutation Graphs

Polar graphs generalise bipartite, cobipartite, split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.KeywordsInput GraphChordal GraphGraph ClassSplit GraphPermutation GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Kneser graphs are like Swiss cheese, Discrete Analysis 2018:2, 18 pp. This paper relates two very interesting areas of research in extremal combinatorics: removal lemmas, and of variables. A _removal lemma_ is a result that states that if $S$ is a set that contains only few copies of a certain structure, then one can remove just a few elements from $S$ to create a set $S'$ that has no copies of that structure. For instance, the triangle removal states that a graph with few triangles can be approximated by a graph with no triangles. Removal lemmas play a very important role in extremal and additive combinatorics. As for the of variables, the aspect that is of interest here is that there are many interesting theorems that state that if a function has a certain property, it must be because in a certain sense it depends on only a few variables. For example, a famous result of the first author shows that if a monotone graph property fails to have a sharp threshold, then the property can be approximated by one for which the minimal graphs that satisfy it are all small, with the result that the influence of individual edges on whether the graph has the property is substantial. The main result of this paper is an removal lemma for various families of graphs. It states that if $G$ is a graph in one of these families, then for every set $X$ of vertices in $G$ that spans only a few edges one can throw away just a small proportion of $X$ to obtain a subset $X'$ that is independent. Of course, a result like this is completely false in general: one has only to take a sparse random graph, and then every subset will span few edges but no large subset will be independent. But it is true, for example, for complete bipartite graphs, since a set of vertices that spans a sparse subgraph will have to be contained almost entirely in one of the two vertex sets, and then one can throw away the vertices that are in the other set. One of the families for which the authors prove an edge-removal is that of Kneser graphs. For $0<m<n/2$, the Kneser graph $K(n,m)$ has as its vertex set the set of all subsets of $\{1,2,\dots,n\}$ of size $m$, with two such subsets joined by an edge if and only if they are disjoint. Kneser graphs have large sparse sets, but the natural examples seem to depend on just a few variables: for example, one can take all sets that contain a particular element. (Note that an independent set in $K(n,m)$ is just an intersecting family of $m$-sets.) The authors wanted to prove a precise conjecture that would say that every set of vertices that spans a sparse subgraph can be approximated by a set of vertices that (i) depends on just a few variables and (ii) spans an independent set. Before this paper a weaker statement was known where instead of (ii) one had a sparse set. In this paper the authors settle the conjecture completely by showing that a sparse set of vertices can be approximated by an independent set of vertices. The other family of graphs for which the authors prove an edge removal is that of product graphs. If $H$ is a graph, then the product $G=H^n$ is the graph where $x$ and $y$ are joined if for every coordinate $i$ we have that $x_iy_i$ is an edge of $H$. (There are different definitions of product graphs, but this is the one to which the result applies: note that it makes it quite hard for two vertices to be joined.) For example, if $H$ is a complete graph on $r$ coordinates, then $G=[r]^n$ with two vertices joined if and only if they differ in every coordinate. The proof of the removal uses ideas from Jacob Fox's proof of the triangle removal lemma, which obtains the best known bound for that result. Finally, what has this to do with Swiss cheese? The rough idea is that these graphs have large and highly structured independent sets, and a part of the graph is sparse if and only if it has a big intersection with one of these independent sets. Likewise a Swiss cheese contains lots of large holes, and a region of space contained within the boundary of the cheese will contain only a small amount of cheese if and only if it is almost entirely contained in one of the holes. Article image [by G. J. Wanderer](https://www.flickr.com/photos/106561245@N07/13515457313/in/photolist-mAjfA6-f7mF8i-Tf5Ler-7i4FtY-bi74Nt-gWVXdj-7XoC4N-okRvp5-77HxiZ-7xt4hx-RSUt3B-qeB17D-as2aAc-7K64uq-oL2GMj-6trVsK-UDq452-SP1xSX-d4u57o-fATSUi-8Km2Qs-dJeLzb-qu9Z4Z-dNNvQ2-7eVY4b-6mtCks-frWJEF-aCerNx-7tyjAW-haiMTG-oysrog-9LJVj8-fzno-4VuER1-b5n85M-Jka6K5-a7vHJZ-5S7Yyt-Uwa6kC-dNU7Mm-5AWGod-YSE7a-bzagV7-8tV58z-8YW1XL-5nvECU-QAdC69-8wG84t-TSsHP4-9eoj6S) and distributed under a [CC-BY-NC-SA licence](https://creativecommons.org/licenses/by-nc-sa/2.0/).

The Unitary addition Cayley graph of Gaussian integers modulo n denoted by \(G_n[i]\) has a vertex set \(\mathbb {Z}_n[i]\). Any two vertices, \(x=a_1+ib_1\), \(y=a_2+ib_2\) of \(G_n[i]\), are adjacent if and only if \(gcd(N(x+y),n)=1\), where \(N(a+ib)=a^2+b^2\) is the norm of any element \(a+ib\) in \(\mathbb {Z}_n[i]\). First, we find two important results of unitary addition Cayley graph of Gaussian integers modulo n by considering \(\mathbb {Z}_n[i]\) and \(U_n[i]\) as the vertex set. Using the results, we obtain clique number, chromatic number of the graph \(G_n[i]\). Lastly, domination number of unitary addition Cayley graph of Gaussian integers modulo n is also found in this paper.KeywordsCayley graphGaussian integersClique numberDomination number

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set Ic of all countable graphs (since every graph in Ic is isomorphic to an induced subgraph of R). A brief overview of known universality results for some induced-hereditary subsets of Ic is provided. We then construct a k-degenerate graph which is universal for the induced-hereditary property of finite k-degenerate graphs. In order to attempt the corresponding problem for the property of countable graphs with colouring number at most k + 1, the notion of a property with assignment is introduced and studied. Using this notion, we

Polynomial-time algorithms for the subset feedback vertex set problem on interval graphs and permutation graphs

Let 𝑋 be a set and (𝑋,𝑇) be a topological space. A new type of graph on 𝑃(𝑋), namely the closure graph of 𝑇 is introduced. The closure graph denoted by &lt;em&gt;Γ&lt;sup&gt;c&lt;/sup&gt;&lt;/em&gt; whose vertex set is 𝑃(𝑋) in which two distinct vertices 𝐴 and 𝐵 are adjacent if &lt;em&gt;A&lt;/em&gt;'∩&lt;em&gt;B&lt;/em&gt;' ⊆ (&lt;em&gt;A&lt;/em&gt;∩&lt;em&gt;B&lt;/em&gt;)'. In this paper, the closure graph is shown as a simple, connected graph with diameter at most two. Furthermore, the girth of the closure graph &lt;em&gt;Γ&lt;sup&gt;c&lt;/sup&gt;&lt;/em&gt;of 𝑇 is three if 𝑋 contains more than one point. Also, several graph properties are studied.

A set of vertices and edges forms a graph. A graph can be associated with groups using the groups' properties for its vertices and edges. The set of vertices of the graph comprises the elements of the group, while the set of edges of the graph is the properties and requirements for the graph. A non-abelian tensor square graph of a group is defined when its vertex set represents the non-tensor centre elements' set of G. Then, two distinguished vertices are connected by an edge if and only if the non-abelian tensor square of these two elements is not equal to the identity of the non-abelian tensor square. This study investigates the non-abelian tensor square graph for the symmetric group of order six. In addition, some properties of this group's non-abelian tensor square graph are computed, including the diameter, the dominating number, and the chromatic number. The perfect code for the non-abelian tensor square graph for a symmetric group of order six is also found in this paper.

Finding a central vertex in an HHD-free graph

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] g and [y,x] g. We denote this graph by . In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number and dominating number. We prove that if G, H are isoclinic groups with |Z (G)|=|Z (H)|, then their associated graphs are isomorphic.

In the constraint satisfaction problem CSP(H) corresponding to a finite relational structure H, the aim is to decide, given a relational structure G, whether there exists a homomorphism from G to H. In (Bulatov, 2003), we proved that if H is a conservative structure, then it can be associated with a complete edge-3-colored graph whose vertex set is the universe of H. The complexity and a solution algorithm for CSP(H) strongly depend on certain properties of the associated graph. In this paper we show how a similar edge-3-colored graph can be defined for an arbitrary finite relational structure H. Then we study properties of the defined graph and find a solution algorithm for CSP(H), where G(H) satisfies some restrictions. The latter result substantially generalizes the results (2000,2002,1998,1997) concerning max-closed constraints and constraints with a 2-semilattice, semigroup or conservative groupoid polymorphism. Finally, we complete the study of the complexity of maximal constraint languages started in (Bulatov et al., 2001).

The integral circulant graph $X_n (D)$ has the vertex set $Z_n = \{0, 1,\ldots$, $n{-}1\}$ and vertices $a$ and $b$ are adjacent, if and only if $\gcd(a{-}b$, $n)\in D$, where $D = \{d_1,d_2, \ldots, d_k\}$ is a set of divisors of $n$. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph $X_n (1)$ and the disconnected graph $X_n (d)$. In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups $Aut (X_n (1, p))$ for $n$ being a square-free number and $p$ a prime dividing $n$, and $Aut (X_n (1, p^k))$ for $n$ being a prime power.

An algorithm for finding clique cut-sets

&lt;p&gt;For standard terminology and notion in graph theory we refer the reader to Harary; the&nbsp;non-standard will be given in this paper as and when required. We treat only finite simple&nbsp;graphs without self loops and isolates.&lt;/p&gt;

Let R be a ring with unity and I(R)∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R)∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Zn)). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Zn)).

In this paper, a new kind of graph on a finite group , namely the result involution graph is defined and studied. We use to denote this graph, is a simple undirected graph with vertex set. Two distinct vertices are adjacent if and only if their product is nontrivial involution element in . The result involution graph for several finite groups are obtained. We study some properties of the result involution graph by resizing graph by using the conjugacy classes of . Finally, we show that the result involution graphs for the symmetric groups and the alternating groups are connected with diameter at most 3 and radius at most 2 for. Furthermore, they have girth 3.