Two characterizations of chain partitioned probe graphs

Chain graphs and threshold graphs are special classes of graphs which have maximum spectral radius among bipartite graphs and connected graphs with given order and size, respectively. In this article, we focus on some of linear algebraic tools like rank, determinant, and permanent related to the adjacency matrix of these types of graphs. We derive results relating the rank and number of edges. We also characterize chain/threshold graphs with nonzero determinant and permanent.

Polar cographs

Let G be a simple connected molecular graph with vertex set $V(G)$ and edge set $E(G)$. One important modification of classical Zagreb index, called hyper Zagreb index $HM(G)$ is defined as the sum of squares of the degree sum of the adjacent vertices, that is, sum of the terms $[{d_G}(u)+{d_G}(v)]^2$ over all the edges of $G$, where ${d_G}(u)$ denote the degree of the vertex $u$ of $G$. In this paper, the hyper Zagreb index of certain bridge and chain graphs are computed and hence using the derived results we compute the hyper Zagreb index of several classes of chemical graphs and nanostructures.

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/).

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent set. We present properties of different-distance sets, and consider the different-distance numbers of paths, cycles, Cartesian products of bipartite graphs, and Cartesian products of complete graphs. We conclude with some open problems and questions.

Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.

The chain graph sandwich problem asks: Given a vertex set V, a mandatory edge set E 1, and a larger edge set E 2, is there a graph G=(V,E) such that E 1⊆E⊆E 2 with G being a chain graph (i.e., a 2K 2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E 1 is a connected graph, which has a linear-time solution, (2) the threshold graph sandwich problem, which has a linear-time solution, and (3) the chain probe graph problem, which has a polynomial-time solution.

An independent set $I_c $ of a undirected graph G is called critical if \[|I_c | - |N( I_c )| = \max \{ |I| - |N( I )|:I\,{\text{is an independent set of }}G \},\] where $N( I )$ is the set of all vertices of G adjacent to some vertex of I. It has been proved by Cun-Quan Zhang [SIAM J. Discrete Math., 3 (1990), pp. 431–438] that the problem of finding a critical independent set is polynomially solvable. This paper shows that the problem can be solved in $O( |V ( G )|^{1/2} |E ( G )| )$ time and its weighted version in $O( |V ( G )|^2 |E ( G )|^{1/2} )$ time.

A graph G is said to be a bisplit graph if its vertex set can be partitioned into a stable set and a complete bipartite graph. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph G and a subset of vertices \(R\subseteq V(G)\), the objective is to find a minimum cardinality set \(S\subset V (G)\) such that the set \(R \cup S\) induces a connected subgraph. In this paper, we present an interesting dichotomy result for STREE on bisplit graphs, we show that STREE is polynomial-time solvable for chordal bipartite bisplit graphs, and NP-complete otherwise. Further, we show that for chordal bisplit graphs, the problem is polynomial-time solvable. A revisit of our NP-complete reduction instances reveals that the instances are diameter at most 5 bipartite graphs. We also obtain one more dichotomy result for STREE on bisplit graphs which says that for diameter 5 the problem is NP-complete and polynomial-time solvable for diameter at most 4. On the parameterized complexity front, we show that the parameterized version of Steiner tree problem on bisplit graphs is fixed-parameter tractable when the parameter is the biclique size and is W[2]-hard on bisplit graphs if the parameter is the solution size. We conclude this paper by presenting structural results of bisplit graphs which will be of use to solve other combinatorial problems.KeywordsBisplit graphsThe Steiner tree problemW-hardnessDichotomy results

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.

Clustering and domination in perfect graphs

A pair of nonnegative integer sequences {D1, D2} with D1 = (d1,1, d1,2, …, dnD2= (d2,1, d2,2, …, dna bipartite graphical sequence, if there is a bipartite graph G with degrees{D1, D2} (i.e., G has two independent vertex sets V1 = {v1,1, v1,2, …, vnV2 = {v2,1, v2,2, …, vnsuch that di,ji the degree of vertex vi,jiG for each i = 1, 2 and ji = 1, 2, …, ni). In other words,{D1, D2} is a bipartite graphical sequence if and only if there is an n1× n2 matrix of 0's and 1's havingdjj1 and djj2. The connectivity κ({D1, D2}) of a bipartite graphical sequence{D1, D2} is defined to be the maximum integer k such that there is a k-connected bipartite graph with degrees {D1, D2}. In this paper, we present a characterization of a k-connected bipartite graphical sequence and an O(n log log n) time algorithm, for a given bipartite graphical sequence {D1, D2}, to construct a κ({D1, D2})-connected bipartite graph with degrees {D1, D2} (n = n1+n2). © 1997 John Wiley & Sons, Inc.

Approximating permanents and hafnians, Discrete Analysis 2017:2, 34 pp. The _permanent_ per$(A)$ of an $n\times n$ matrix $A$ is like the determinant but without the sign changes: that is, it is the sum $\sum_{\pi\in S_n}\prod_{i=1}^nA_{i\pi(i)}$. If $A$ is the bipartite adjacency matrix of a bipartite graph $G$, then per$(A)$ is the number of perfect matchings in $G$. Whereas row-operations allow one to compute the determinant of a matrix efficiently, computing the permanent is known, by a famous result of Valiant, to be hard (in the sense that if you could compute it efficiently, then you would be able to give efficient solutions to a wide variety of enumeration problems, and would in particular have shown that P=NP). However, another famous result, due to Jerrum, Sinclair and Vigoda, which built on earlier work of Jerrum and Sinclair, shows that there are probabilistic algorithms that run in polynomial time and approximate the permanent to within a factor $1+\epsilon$ with high probability. This paper concerns the question of how well the permanent can be approximated using a _deterministic_ algorithm. The main result of this paper is that if $\delta,\epsilon>0$ then for each $n\in\mathbb N$ there is a polynomial $p$ in $n^2$ variables of degree $O_\delta(\log n+\log(1/\epsilon))$ such that if $A$ is any $n\times n$ matrix with all its entries between $\delta$ and 1, then the permanent of $A$ can be approximated to within a factor $1+\epsilon$ by $\exp(p(A))$. Here, of course, $p(A)$ is $p$ applied to the entries of $A$. Furthermore, $p(A)$ can be computed in time $n^{O_\delta(\log n+\log(1/\epsilon))}$. As mentioned above, the permanent of a 01-matrix counts the number of perfect matchings in a bipartite graph. If instead one is interested in the number of perfect matchings in a graph with $2n$ vertices (that is, the number of sets of $n$ disjoint edges), then the corresponding quantity is called the _hafnian_. For a general $2n\times 2n$ matrix $A$, the hafnian haf$(A)$ is the sum over all partitions of $\{1,\dots,2n\}$ into $n$ sets $\{i_r,j_r\}$ of the product $\prod_{r=1}^nA_{i_rj_r}$. The author proves the same result for hafnians that he obtains for permanents. This is interesting, because it is not known how to approximate hafnians even with a randomized algorithm. The permanent and hafnian are polynomials of degree $n$ in the matrix entries, so these results say that on the class of matrices being considered (that is, real matrices with all entries between $\delta$ and 1), they can be approximated by polynomials of much lower degree. The proofs depend on results that say that the two polynomials cannot be zero on a complex matrix that is close to this class. Similar results are proved for complex matrices with all their entries close to 1. In this case the methods carry over to permanents of multidimensional tensors, but multidimensional generalizations appear to be harder to obtain for the results mentioned above, for reasons explained in the paper, and are not currently known.

The expansion $G^+$ of a graph $G$ is the 3-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let ${ex}_3(n,F)$ denote the maximum number of edges in a 3-uniform hypergraph with $n$ vertices not containing any copy of a 3-uniform hypergraph $F$. The study of ${ex}_3(n,G^+)$ includes some well-researched problems, including the case that $F$ consists of $k$ disjoint edges, $G$ is a triangle, $G$ is a path or cycle, and $G$ is a tree. In this paper we initiate a broader study of the behavior of ${ex}_3(n,G^+)$. Specifically, we show $ {ex}_3(n,K_{s,t}^+) = \Theta(n^{3 - 3/s})$ whenever $t > (s - 1)!$ and $s \geq 3$. One of the main open problems is to determine for which graphs $G$ the quantity ${ex}_3(n,G^+)$ is quadratic in $n$. We show that this occurs when $G$ is any bipartite graph with Turán number $o(n^{\varphi})$ where $\varphi = \frac{1 + \sqrt{5}}{2}$, and in particular this shows ${ex}_3(n,G^+) = O(n^2)$ when $G$ is the three-dimensional cube graph.

Bipartite bithreshold graphs