Polarity on H-split graphs

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

A graph is called a split graph if its vertex set can be partitioned into a clique and an independent set. Split graphs have rich mathematical structure and interesting algorithmic properties making it one of the most well-studied special graph classes. In the Split Vertex Deletion(SVD) problem, given a graph and a positive integer k, the objective is to test whether there exists a subset of at most k vertices whose deletion results in a split graph. In this paper, we design a kernel for this problem with \(\mathcal {O}(k^2)\) vertices, improving upon the previous cubic bound known. Also, by giving a simple reduction from the Vertex Cover problem, we establish that SVD does not admit a kernel with \(\mathcal {O}(k^{2-\epsilon })\) edges, for any \(\epsilon >0\), unless \(\textsf {NP}\subseteq \textsf {coNP/poly}\).

The noncommuting graph $nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.

A 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 γ is (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.

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

Complete Multipartite Graphs and their Null Set

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/).

On this research we study EPG graphs, in particular, we are interested on investigate the intersection between split and B2-EPGgraph classes. The results found in the literature regarding split graphs only concern about characterization in B1-EPG, that despite bringing some graphs that belong to this class, the bend number of split graphs it is still unknown. We study the split graphs whose degree of vertices on independent set is less than 2. In this work we manipulate bipartite graphs and we present representation for some bipartite graphs in B2-EPG. We build an algorithm that create a Split B2-EPG representation, also we present another algorithm that builds a representation on a grid Qw×2y+1 for any Split graph, and other results.

Edge vulnerability parameters of split graphs

A graph $G = (C, I, E)$ is called a split graph if its vertex set $V$ can be partitioned into a clique $C$ and an independent set $I$. A graph $G = (Y \cup Z, I, E)$ is called a bisplit graph if its vertex set $V$ can be partitioned into three stable sets $I, Y,Z$ such that $Y \cup Z$ induces a complete bipartite graph and an independent set $I$. A connected graph $G$ is called supper-$κ$ (resp. super-$λ$) if every minimum vertex cut (edge cut) of $G$ is the set of neighbors of some vertex (the edges of incident to some vertex) in $G$. In this note, we show that: split graphs and bisplit graphs are super-$κ$ and super-$λ$.

For a graph G, a set D⊆V(G) is called a [1,j]-dominating set if every vertex in V(G)∖D has at least one and at most j neighbors in D. A set D⊆V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study the aforementioned problems on various graph classes from the perspective of parameterized complexity and prove the following results:•[1,j]-Dominating Set parameterized by solution size is W[1]-hard on d-degenerate graphs for d=j+1.•[1,j]-Dominating Set parameterized by solution size is FPT on nowhere dense graphs.•The known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH).•Assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth.•Finally, we study another variant of Dominating Set, called Restrained Dominating Set, that asks if there is a dominating set D of G of size at most k such that no vertex outside of D has all of its neighbors in D. We prove this variant is W[1]-hard even on 3-degenerate graphs.

On Semi-Transitive Orientability of Split Graphs

The last chapter of this book eventually presents some famous classes of perfect graphs, namely comparability, interval, permutation, and split graphs. We first present an example of a graph which is at the same time a triangulated, a comparability, a split, and a permutation graph. The importance to be an interval is illustrated with Berge’s mystery story. We discuss furthermore the generation of permutation graphs and close with how to recognise that a given graph is in fact a permutation graph.

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.