The three well-known graph classes, planar graphs (P), series-parallel graphs(SP) and outer planar graphs(OP) satisfy the following proper inclusion relation: OP C SP C P. It is known that box(G) <= 3 if G belongs to P and box(G) <= 2 if G belongs to OP. Thus it is interesting to decide whether the maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In this paper we construct a series-parallel graph with boxicity 3, thus resolving this question. Recently Chandran and Sivadasan showed that for any G, box(G) <= treewidth(G)+2. They conjecture that for any k, there exists a k-tree with boxicity k+1. (This would show that their upper bound is tight but for an additive factor of 1, since the treewidth of any k-tree equals k.) The series-parallel graph we construct in this paper is a 2-tree with boxicity 3 and is thus a first step towards proving their conjecture.
The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.
For a graph $G$, its \emph{cubicity} $cub(G)$ is the minimum dimension $k$ such that $G$ is representable as the intersection graph of (axis--parallel) cubes in $k$--dimensional space. Chandran, Mannino and Oriolo showed that for a $d$--dimensional hypercube $H_d$, $\frac{d-1}{\log d} \le cub(H_d) \le 2d$. In this paper, we show that $cub(H_d) = Θ(\frac{d}{\log d})$.The parameter \emph{boxicity} generalizes cubicity: the boxicity $box(G)$ of a graph $G$ is defined as the minimum dimension $k$ such that $G$ is representable as the intersection graph of axis parallel boxes in $k$ dimensional spa…
An axis-parallel $d$--dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_d$ where $R_i$ (for $1 \le i \le d$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its \emph{boxicity} $\boxi(G)$ is the minimum dimension $d$, such that $G$ is representable as the intersection graph of (axis--parallel) boxes in $d$--dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. We show that for any graph $G$ with maximum degree $Δ$, $\boxi(G) \le 2 Δ^2 + 2$. That the bound does not depend …
An axis-parallel b-dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_b$ where each $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form $[a_i,b_i]$ on the real line. The boxicity of any graph $G$, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product $R_1 \times R_2\times ... \times R_b$, where each $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form [$a_i$,$a_i$+1] on the real line. When the boxes are restricted to be axis-…
The boxicity of a graph G, denoted as box(G) is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are a subclass of strongly chordal graphs and are used in the construction of phylogenetic trees in evolutionary biology. We show that for a k-leaf power G, box(G)\leq k-1. We also show the tightness …
The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of $d$, of the boxicity and the cubicity of the $d$-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubici…
An axis-parallel $b$-dimensional box is a Cartesian product $R_1\times R_2\times...\times R_b$ where $R_i$ is a closed interval of the form $[a_i,b_i]$ on the real line. For a graph $G$, its \emph{boxicity} box(G) is the minimum dimension $b$, such that $G$ is representable as the intersection graph of boxes in $b$-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which …
An axis-parallel $d$-dimensional box is a cartesian product $I_1\times I_2\times \dots \times I_b$ where $I_i$ is a closed sub-interval of the real line. For a graph $G = (V,E)$, the $boxicity \ of \ G$, denoted by $\text{box}(G)$, is the minimum dimension $d$ such that $G$ is the intersection graph of a family $(B_v)_{v\in V}$ of $d$-dimensional boxes in $\mathbb R^d$. Let $k$ and $n$ be two positive integers such that $n\geq 2k+1$. The $Kneser \ graph$ $Kn(k,n)$ is the graph with vertex set given by all subsets of $\{1,2,\dots,n\}$ of size $k$ where two vertices are adjacent if their corresp…
The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 α\right \rceil$, where $α= α(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 χ(G) \right \rceil box(G) …
An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denot…
Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even for graph classes like bipartite, co-bipartite and split graphs within $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. We give FPT approximation algorithms for computing the boxicity of graphs, where the parameter used is the vertex or edge edit distance of the given graph from families of graphs of bounded boxicity. This can be seen …
Boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper, we show that for a line graph G of a multigraph, box(G) <= 2Δ(\lceil log_2(log_2(Δ)) \rceil + 3) + 1, where Δdenotes the maximum degree of G. Since Δ<= 2(χ- 1), for any line graph G with chromatic number χ, box(G) = O(χlog_2(log_2(χ))). For the d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d)) \rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d) was left open by Chandran and Siva…
Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^…
By means of analytic techniques we show that the expected number of spanning trees in a connected labelled series-parallel graph on $n$ vertices chosen uniformly at random satisfies an estimate of the form $s \varrho^{-n} (1+o(1))$, where $s$ and $\varrho$ are computable constants, the values of which are approximately $s \approx 0.09063$ and $\varrho^{-1} \approx 2.08415$. We obtain analogue results for subfamilies of series-parallel graphs including 2-connected series-parallel graphs, 2-trees, and series-parallel graphs with fixed excess.
Bouchet conjectured in 1983 that each signed graph that admits a nowhere-zero flow has a nowhere-zero 6-flow. We prove that the conjecture is true for all signed series-parallel graphs. Unlike the unsigned case, the restriction to series-parallel graphs is nontrivial; in fact, the result is tight for infinitely many graphs.
In this paper we give an algorithm to determine, for any given suborder closed class of series-parallel posets, a structure theorem for the class. We refer to these structure theorems as structural descriptions.
The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.
We prove that the maximum degree $Δ_n$ of a random series-parallel graph with $n$ vertices satisfies $Δ_n/\log n \to c$ in probability, and $\mathbb{E}\, Δ_n \sim c \log n$ for a computable constant $c>0$. The same result holds for outerplanar graphs.
In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain $K_4$ as a minor. Series-parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present a proof that every connected series-parallel graph has a vertex that is co…
We show that the number $g_n$ of labelled series-parallel graphs on $n$ vertices is asymptotically $g_n \sim g\cdot n^{-5/2} γ^n n!$, where $γ$ and $g$ are explicit computable constants. We show that the number of edges in random series-parallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs and for graphs not containing $K_{2,3}$ as a minor.
We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths.
Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement of the Tamari one. Moreover, we provide a description of both the weak and the strong Bruhat order on 312-avoiding permutations, by recovering the proof of the fact that they are isomorphic to the Tamari and the Dyck order, respectively; our proof, wh…
We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce gene…
The strong fractional choice number of a graph $G$ is the infimum of those real numbers $r$ such that $G$ is $(\lceil rm \rceil, m)$-choosable for every positive integer $m$. The strong fractional choice number of a family ${\cal G}$ of graphs is the supremum of the strong fractional choice number of graphs in ${\cal G}$. We denote by ${\cal{Q}}_k$ the class of series-parallel graphs with girth at least $k$. This paper proves that for $k=4q-1, 4q,4q+1, 4q+2$, the strong fractional number of ${\cal{Q}}_k$ is exactly $2+ \frac{1}{q}$.
We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.
We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow Lambda lie in the disc |q-1| < (Lambda-1)/log 2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the "real antiferromagnetic regime" -1 \le v_e \le 0. This result is within a factor 1/log 2 \approx 1.442695 of being sharp
Let $(h_0,h_1,\ldots,h_s)$ with $h_s\ne0$ be the $h$-vector of the broken circuit complex of a series-parallel network $M$. Let $G$ be a graph whose cycle matroid is $M$. We give a formula for the difference $h_{s-1}-h_1$ in terms of an ear decomposition of $G$. A number of applications of this formula are provided, including several bounds for $h_{s-1}-h_1$, a characterization of outerplanar graphs, and a solution to a conjecture on $A$-graphs posed by Fenton. We also prove that $h_{s-2}\geq h_2$ when $s\geq 4$.
The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes …
School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）；School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）；Qinghai Institute of Applied Mathematics（青海省应用数学研究所）；School of Mathematics and System Sciences, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）
The Cops and Robber game is played on undirected finite graphs. $k$ cops and one robber are positioned on vertices and take turn in moving along edges. The cops win if, after a move, a cop and the robber are on the same vertex. A graph is called $k$-copwin, if the cops have a winning strategy. It is known that planar graphs are 3-copwin (Aigner & Fromme, 1984) and that outerplanar graphs are 2-copwin (Clarke, 2002). In this short note, we prove that series-parallel (i.e., graphs with no $K_4$ minor) graphs are 2-copwin. It is a well-known trick in the literature of cops & robber games to defin…
The series-parallel (SP) graphs are those containing no topological $K_{_4}$ and are considered trivial. We relax the prohibition distinguishing the SP graphs by forbidding only embeddings of $K_{_4}$ whose edges with both ends 3-valent (skeleton hereafter) induce a graph isomorphic to certain prescribed subgraphs of $K_{_4}$. In particular, we describe the structure of the graphs containing no embedding of $K_{_4}$ whose skeleton is isomorphic to $P_{_3}$ or $P_{_4}$. Such "almost series-parallel graphs" (ASP) still admit a concise description. Amongst other things, their description reveals …
We give a simpler proof of Seymour's Theorem on edge-coloring series-parallel multigraphs and derive a linear-time algorithm to check whether a given series-parallel multigraph can be colored with a given number of colors.
We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.
In 1980, White conjectured that the toric ideal associated to a matroid is generated by binomials corresponding to a symmetric exchange. In this paper, we prove that classes of matroids for which the toric ideal is generated by quadrics and that has quadratic Gröbner bases, are closed under series and parallel extensions, series and parallel connections, and 2-sums.
A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most $\lfloor 5n/8 \rfloor$ paths, which is an improvement over the best previously known bound of $\lfloor 2n/3 \rfloor$.
Bandelt and Mulder's structural characterization of Bipartite Distance Hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pending vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin's structural characterization of 2-connected series-parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we prove that the two constructions are the same construction when bipartite graphs…
Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $δ(U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $Ω$ such that $w(Ω)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the…
The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids: oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyh…
We consider the identifiability problem for the parameters of series-parallel LCR circuit networks. We prove that for networks with only two classes of components (inductor-capacitor (LC), inductor-resistor (LR), and capacitor-resistor (RC)), the parameters are identifiable if and only if the number of non-monic coefficients of the constitutive equations equals the number of parameters. The notion of the "type" of the constitutive equations plays a key role in the identifiability of LC, LR, and RC networks. We also investigate the general series-parallel LCR circuits (with all three classes of…
We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time $O(mΔ^3)$, where $Δ$ is the maximum degree of the graph with $m$ edges. We use $O(mΔ^2 \log Δ)$ bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result that all constructible tropical linear spaces have the same f-vector and are ``series-parallel''. We conjecture that this f-vector is maximal for all tropical linear spaces with equality precisely for the series-parallel tropical linear spaces. We present many partial results towards this co…
The series-parallel (active) redundancy allocation problem with mixed components (RAP) involves setting reliable objectives for components or subsystems to meet the resource consumption constraint, e.g., the total cost. RAP has been an active research area for the past four decades. The NP-hard difficulties confronted by RAP are maintaining feasibility with respect to two constraints: cost and weight. A novel algorithm called the bound-rule-BAT (BRB) based on the binary-addition-tree algorithm (BAT), the dominance rule, and dynamic bounds are proposed to solve the exact solutions of the most f…
