Clique Cover Width and Clique Sum

For a clique cover $C$ in the undirected graph $G$, the clique cover graph of $C$ is the graph obtained by contracting the vertices of each clique in $C$ into a single vertex. The clique cover width of G, denoted by $CCW(G)$, is the minimum value of the bandwidth of all clique cover graphs of $G$. When $G$ is the clique sum of $G_1$ and $G_2$, we prove that $CCW(G) \le 3/2(CCW(G_1) + CCW(G_2))$

New separation theorems and sub-exponential time algorithms for packing and piercing of fat objects

For $\cal C$ a collection of $n$ objects in $R^d$, let the packing and piercing numbers of $\cal C$, denoted by $Pack({\cal C})$, and $Pierce({\cal C})$, respectively, be the largest number of pairwise disjoint objects in ${\cal C}$, and the smallest number of points in $R^d$ that are common to all elements of ${\cal C}$, respectively. When elements of $\cal C$ are fat objects of arbitrary sizes, we derive sub-exponential time algorithms for the NP-hard problems of computing ${Pack}({\cal C})$ and $Pierce({\cal C})$, respectively, that run in $n^{O_d({{Pack}({\cal C})}^{d-1\over d})}$ and $n^{O_d({{Pierce}({\cal C})}^{d-1\over d})}$ time, respectively, and $O(n\log n)$ storage. Our main tool which is interesting in its own way, is a new separation theorem. The algorithms readily give rise to polynomial time approximation schemes (PTAS) that run in $n^{O({({1\over\epsilon})}^{d-1})}$ time and $O(n\log n)$ storage. The results favorably compare with many related best known results. Specifically, our separation theorem significantly improves the splitting ratio of the previous result of Chan, whereas, the sub-exponential time algorithms significantly improve upon the running times of very recent algorithms of Fox and Pach for packing of spheres.Comment: 28th European Workshop on Computational Geometry,2012 - Assisi, Perugia, Italy, 269-27

New representation results for planar graphs

A universal representation theorem is derived that shows any graph is the intersection graph of one chordal graph, a number of co-bipartite graphs, and one unit interval graph. Central to the the result is the notion of the clique cover width which is a generalization of the bandwidth parameter. Specifically, we show that any planar graph is the intersection graph of one chordal graph, four co-bipartite graphs, and one unit interval graph. Equivalently, any planar graph is the intersection graph of a chordal graph and a graph that has {clique cover width} of at most seven. We further describe the extensions of the results to graphs drawn on surfaces and graphs excluding a minor of crossing number of at most one.Comment: 29th European Workshop on Computational Geometry March 17-20, 2013, 177-18

Unit Incomparability Dimension and Clique Cover Width in Graphs

For a clique cover $C$ in the undirected graph $G$, the {\it clique cover graph} of $C$ is the graph obtained by contracting the vertices of each clique in $C$ into a single vertex. The {\it clique cover width} of $G$, denoted by $CCW(G)$, is the minimum value of the bandwidth of all clique cover graphs in $G$. Any $G$ with $CCW(G)=1$ is known to be an incomparability graph, and hence is called, a {\it unit incomparability graph}. We introduced the {\it unit incomparability dimension of $G$}, denoted by$Udim(G)$, to be the smallest integer $d$ so that there are unit incomparability graphs $H_i$ with $V(H_i)=V(G), i=1,2,...,d$, so that $E(G)=\cap_{i=1}^d E(G_i)$. We prove a decomposition theorem establishing the inequality $Udim(G)\le CCW(G)$. Specifically, given any $G$, there are unit incomparability graphs $H_1,H_2,...,H_{CC(W)}$ with $V(H_i)=V(G)$ so that and $E(G)=\cap_{i=1}^{CCW} E(H_i)$. In addition, $H_i$ is co-bipartite, for $i=1,2,...,CCW(G)-1$. Furthermore, we observe that $CCW(G)\ge s(G)/2-1$, where $s(G)$ is the number of leaves in a largest induced star of $G$ , and use Ramsey Theory to give an upper bound on $s(G)$, when $G$ is represented as an intersection graph using our decomposition theorem. Finally, when $G$ is an incomparability graph we prove that $CCW (G)\le s(G)-1$

Bounds for the Clique Cover Width of Factors of the Apex Graph of the Planar Grid

The {\it clique cover width} of $G$, denoted by $ccw(G)$, is the minimum value of the bandwidth of all graphs that are obtained by contracting the cliques in a clique cover of $G$ into a single vertex. For $i=1,2,...,d,$ let $G_i$ be a graph with $V(G_i)=V$, and let $G$ be a graph with $V(G)=V$ and $E(G)=\cap_{i=1}^d(G_i)$, then we write $G=\cap_{i=1}^dG_i$ and call each $G_i,i=1,2,...,d$ a factor of $G$. We are interested in the case where $G_1$ is chordal, and $ccw(G_i),i=2,3...,d$ for each factor $G_i$ is "small". Here we show a negative result. Specifically, let ${\hat G}(k,n)$ be the graph obtained by joining a set of $k$ apex vertices of degree $n^2$ to all vertices of an $n\times n$ grid, and then adding some possible edges among these $k$ vertices. We prove that if ${\hat G}(k,n)=\cap_{i=1}^dG_i$, with $G_1$ being chordal, then, $max_{2\le i\le d}\{ccw(G_i)\}\ge {n^{1\over d-1}\over 2.{(2c)}^{1\over {d-1}}}$, where $c$ is a constant. Furthermore, for $d=2$, we construct a chordal graph $G_1$ and a graph $G_2$ with $ccw(G_2)\le {n\over 2}+k$ so that ${\hat G}(k,n)=G_1\cap G_2$. Finally, let ${\hat G}$ be the clique sum graph of ${\hat G}(k_i, n_i), i=1,2,...t$, where the underlying grid is $n_i\times n_i$ and the sum is taken at apex vertices. Then, we show ${\hat G}=G_1\cap G_2$, where, $G_1$ is chordal and $ccw(G_2)\le \sum_{i=1}^t(n_i+k_i)$. The implications and applications of the results are discussed, including addressing a recent question of David Wood

Largest reduced neighborhood clique cover number revisited

Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hat\beta}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hat\beta}_t(G)$ including the connections to the greatest reduced average density of $G$, or $\bigtriangledown_t(G)$, introduce the class of graphs with bounded neighborhood clique cover number, and derive a simple lower and an upper bound for this important graph parameter. We announce two conjectures, one for the value of ${\hat\beta}_t(G)$, and another for a separator theorem (with respect to a certain measure) for an interesting class of graphs, namely the class of incomparability graphs which we suspect to have a polynomial bounded neighborhood clique cover number, when the size of a largest induced star is bounded.Comment: The results in this paper were presented in 48th Southeastern Conference in Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, March 201

A new upper bound for the clique cover number with applications

Let $\alpha(G)$ and $\beta(G)$, denote the size of a largest independent set and the clique cover number of an undirected graph $G$. Let $H$ be an interval graph with $V(G)=V(H)$ and $E(G)\subseteq E(H)$, and let $\phi(G,H)$ denote the maximum of ${\beta(G[W])\over \alpha(G[W])}$ overall induced subgraphs $G[W]$ of $G$ that are cliques in $H$. The main result of this paper is to prove that for any graph $G$ $${\beta(G)}\le 2 \alpha(H)\phi(G,H)(\log \alpha(H)+1),$$ where, $\alpha(H)$ is the size of a largest independent set in $H$. We further provide a generalization that significantly unifies or improves some past algorithmic and structural results concerning the clique cover number for some well known intersection graphs

A new separation theorem with geometric applications

Let $G=(V(G), E(G))$ be an undirected graph with a measure function $\mu$ assigning non-negative values to subgraphs $H$ so that $\mu(H)$ does not exceed the clique cover number of $H$. When $\mu$ satisfies some additional natural conditions, we study the problem of separating $G$ into two subgraphs, each with a measure of at most $2\mu(G)/3$ by removing a set of vertices that can be covered with a small number of cliques $G$. When $E(G)=E(G_1)\cap E(G_2)$, where $G_1=(V(G_1),E(G_1))$ is a graph with $V(G_1)=V(G)$, and $G_2=(V(G_2), E(G_2))$ is a chordal graph with $V(G_2)=V(G)$, we prove that there is a separator $S$ that can be covered with $O(\sqrt{l\mu(G)})$ cliques in $G$, where $l=l(G,G_1)$ is a parameter similar to the bandwidth, which arises from the linear orderings of cliques covers in $G_1$. The results and the methods are then used to obtain exact and approximate algorithms which significantly improve some of the past results for several well known NP-hard geometric problems. In addition, the methods involve introducing new concepts and hence may be of an independent interest.Comment: Proceedings of EuroCG 2010, Dortmund, Germany, March 22-24, 201

On the largest reduced neighborhood clique cover number of a graph

Let $G$ be a graph and $t\ge 0$. A new graph parameter termed the largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is introduced. Specifically, ${\hat\beta}_t(G)$ is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. We verify that ${\hat\beta}_t(G)=1$ when $G$ is chordal, and, ${\hat\beta}_t(G)\le s$, where $G$ is an incomparability graph that does not have a $t-$shallow minor which is isomorphic to an induced star on $s$ leaves. Moreover, general properties of ${\hat\beta}_t(G)$ including the connections to the greatest reduced average density of $G$, or $\bigtriangledown_t(G)$ are studied and investigated. For instance we show ${{\hat\beta}_t(G)\over 2}\le \bigtriangledown_t(G)\le p.{\hat\beta}_t(G),$ where $p$ is the size of a largest complete graph which is a $t-minor$ of $G$. Additionally we prove that largest ratio of any minimum clique cover to the maximum independent set taken overall $t-$minors of $G$ is a lower bound for ${\hat\beta}_t(G)$. We further introduce the class of bounded neighborhood clique cover number for which ${\hat\beta}_t(G)$ has a finite value for each $t\ge 0$ and verify the membership of geometric intersection graphs of fat objects (with no restrictions on the depth) to this class. The results support the conjecture that the class graphs with polynomial bounded neighborhood clique cover number may have separator theorems with respect to certain measures.Comment: A portion of these results were presented at the 47th Southeast Conference on Combinatorics, Graph Theory and Computing, March 7-11, 2016 and will appear in the conference proceedings, Congressus Numerantium (2016

On the strict endoscopic part of modular Siegel threefolds

In this paper we study the non-holomorphic strict endoscopic parts of inner cohomology spaces of a modular Siegel threefold respect to local systems. First we show that there is a non-zero subspace of the strict endoscopic part such that it is constructed by global theta lift of automorphic froms of $(GL(2)\times GL(2))/G_{m}$. Secondly, we present an explicit analytic calculation of levels of lifted forms into GSp(4), based on the paramodular representations theory for $GSp(4; F)$. Finally, we prove the conjecture, by C. Faber and G. van der Geer, that gives a description of the strict endoscopic part for Betti cohomology and (real) Hodge structures in the category of mixed Hodge structures, in which the modular Siegel threefold has level structure one