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SIAM J. on Discrete Mathematics

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2008

Volume 22, Issue 4, pp. 1259-1666


A Unified View of Graph Searching

Derek G. Corneil and Richard M. Krueger

SIAM J. Discrete Math. 22, pp. 1259-1276 (18 pages) | Cited 2 times

Online Publication Date: July 25, 2008

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Graph searching is perhaps one of the simplest and most widely used tools in graph algorithms. Despite this, few theoretical results are known about the vertex orderings that can be produced by a specific search algorithm. A simple characterizing property, such as is known for LexBFS, can aid greatly in devising algorithms, writing proofs of correctness, and showing impossibility results. This paper unifies our view of graph search algorithms by showing simple, closely related characterizations of various well-known search paradigms, including BFS and DFS. Furthermore, these characterizations naturally lead to other search paradigms, namely, maximal neighborhood search and LexDFS.

A Simple Linear Time LexBFS Cograph Recognition Algorithm

Anna Bretscher, Derek Corneil, Michel Habib, and Christophe Paul

SIAM J. Discrete Math. 22, pp. 1277-1296 (20 pages) | Cited 1 time

Online Publication Date: July 25, 2008

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Recently lexicographic breadth first search (LexBFS) has been shown to be a very powerful tool for the development of linear time, easily implementable recognition algorithms for various families of graphs. In this paper, we add to this work by producing a simple two LexBFS sweep algorithm to recognize the family of cographs. This algorithm extends to other related graph families such as $P_4$-reducible, $P_4$-sparse, and distance hereditary. It is an open question whether our cograph recognition algorithm can be extended to a similarly easy algorithm for modular decomposition.

On Computing the Distinguishing Numbers of Planar Graphs and Beyond: A Counting Approach

V. Arvind, Christine T. Cheng, and Nikhil R. Devanur

SIAM J. Discrete Math. 22, pp. 1297-1324 (28 pages) | Cited 1 time

Online Publication Date: September 04, 2008

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A vertex $k$-labeling of graph $G$ is distinguishing if the only automorphism that preserves the labels of $G$ is the identity map. The distinguishing number of $G$, $D(G)$, is the smallest integer $k$ for which $G$ has a distinguishing $k$-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing $k$-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph.

Large Nearly Regular Induced Subgraphs

Noga Alon, Michael Krivelevich, and Benny Sudakov

SIAM J. Discrete Math. 22, pp. 1325-1337 (13 pages)

Online Publication Date: September 04, 2008

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For a real $c \geq 1$ and an integer $n$, let $f(n,c)$ denote the maximum integer $f$ such that every graph on $n$ vertices contains an induced subgraph on at least $f$ vertices in which the maximum degree is at most $c$ times the minimum degree. Thus, in particular, every graph on $n$ vertices contains a regular induced subgraph on at least $f(n,1)$ vertices. The problem of estimating $f(n,1)$ was posed long ago by Erdős, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of $f(n,c)$: (i) For fixed $c>2.1$, $n^{1-O(1/c)} \leq f(n,c) \leq O(cn/\log n)$. (ii) For fixed $c=1+\varepsilon$ with $\varepsilon>0$ sufficiently small, $f(n,c) \geq n^{\Omega(\varepsilon^2/ \ln (1/\varepsilon))}$. (iii) $\Omega (\ln n) \leq f(n,1) \leq O(n^{1/2} \ln^{3/4} n)$. An analogous problem for not necessarily induced subgraphs is briefly considered as well.

Perfect Codes of Length $n$ with Kernels of Dimension $n-\log(n+1)-3$

Olof Heden

SIAM J. Discrete Math. 22, pp. 1338-1350 (13 pages)

Online Publication Date: September 04, 2008

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Perfect 1-error correcting binary codes are considered. Those of length $n$ and with a kernel of dimension $n-\log(n+1)-3$ are shown to be obtainable by the Phelps construction.

Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices

C. Balbuena

SIAM J. Discrete Math. 22, pp. 1351-1363 (13 pages)

Online Publication Date: September 04, 2008

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Let $q$ be a prime power and $r=0,1\ldots, q-3$. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order $q$ by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of $(q-r)$-regular bipartite graphs of girth 6 and $q^2-rq-1$ vertices in each partite set. Moreover, in this work two Latin squares of order $q-1$ with entries belonging to $\{0,1,\ldots, q\}$, not necessarily the same, are defined to be quasi row-disjoint if and only if the Cartesian product of any two rows contains at most one pair $(x,x)$ with $x\ne 0$. Using these quasi row-disjoint Latin squares we find $(q-1)$-regular bipartite graphs of girth 6 with $q^2-q-2$ vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.

Nearly Optimal Visibility Representations of Plane Graphs

Xin He and Huaming Zhang

SIAM J. Discrete Math. 22, pp. 1364-1380 (17 pages)

Online Publication Date: September 11, 2008

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The visibility representation (VR for short) is a classical representation of plane graphs. The VR has various applications and has been extensively studied in the literature. One of the main focuses of the study is to minimize the size of the VR. It is known that there exists a plane graph $G$ with $n$ vertices, where any VR of $G$ requires a size at least $(\lfloor \frac{2n}{3} \rfloor) \times (\lfloor \frac{4n}{3} \rfloor -3)$. For upper bounds, it is known that every plane graph has a VR with height at most $\lfloor \frac{4n-1}{5} \rfloor$, and a VR with width at most $\lfloor \frac{13n-24}{9} \rfloor$. In this paper, we prove that every plane graph has a VR with height at most $\frac{2n}{3}+2\lceil \sqrt{n/2}\rceil$, and a VR with width at most $\frac{4n}{3}+2\lceil \sqrt{n}\rceil$. These representations are nearly optimal in the sense that they differ from the lower bounds only by a lower order additive term. Both representations can be constructed in linear time. Our presentations use Schnyder's realizer to construct the $st$-orientations of plane graphs with special properties. As the $st$-orientation is a very useful concept in other applications, this result may be of independent interest.

Hamilton Cycles in Planar Locally Finite Graphs

Henning Bruhn and Xingxing Yu

SIAM J. Discrete Math. 22, pp. 1381-1392 (12 pages)

Online Publication Date: September 11, 2008

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A classical theorem by Tutte ensures the existence of a Hamilton cycle in every finite $4$-connected planar graph. Extensions of this result to infinite graphs require a suitable concept of an infinite cycle. Such a concept was provided by Diestel and Kühn, who defined circles to be homeomorphic images of the unit circle in the Freudenthal compactification of the (locally finite) graph. With this definition we prove a partial extension of Tutte's result to locally finite graphs.

Prolific Codes with the Identifiable Parent Property

Simon R. Blackburn, Tuvi Etzion, and Siaw-Lynn Ng

SIAM J. Discrete Math. 22, pp. 1393-1410 (18 pages)

Online Publication Date: September 11, 2008

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Let $\cal C$ be a code of length $n$ over an alphabet of size $q$. A word $\mathbf{d}$ is a descendant of a pair of codewords $\mathbf{x},\mathbf{y} \in \cal C$ if $d_i \in \{x_i ,y_i \}$ for $1 \leq i \leq n$. A code $\cal C$ is an identifiable parent property (IPP) code if the following property holds. Whenever we are given $\cal C$ and a descendant $\mathbf{d}$ of a pair of codewords in $\cal C$, it is possible to determine at least one of these codewords. The paper introduces the notion of a prolific IPP code. An IPP code is prolific if all $q^n$ words are descendants. It is shown that linear prolific IPP codes fall into three infinite (“trivial”) families, together with a single sporadic example which is ternary of length $4$. There are no known examples of prolific IPP codes which are not equivalent to a linear example: the paper shows that for most parameters there are no prolific IPP codes, leaving a relatively small number of parameters unsolved. In the process the paper obtains upper bounds on the size of a (not necessarily prolific) IPP code which are better than previously known bounds.

The Independent Even Factor Problem

Satoru Iwata and Kenjiro Takazawa

SIAM J. Discrete Math. 22, pp. 1411-1427 (17 pages)

Online Publication Date: September 11, 2008

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This paper deals with the independent even factor problem. For odd-cycle-symmetric digraphs, in which each arc in any odd dicycle has the reverse arc, a min-max formula is established as a common generalization of the Tutte–Berge formula for matchings and the min-max formula of Edmonds [Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications, R. Guy et al., eds., Gordon and Breach, New York, 1970, pp. 69–87] for matroid intersection. We devise a combinatorial efficient algorithm to find a maximum independent even factor in an odd-cycle-symmetric digraph accompanied by general matroids, which commonly extends two of the alternating-path-type algorithms, the even factor algorithm of Pap [Math. Program., 110 (2007), pp. 57–69], and the matroid intersection algorithms. This algorithm gives a proof of the min-max formula and contains a new operation on matroids, which corresponds to shrinking factor-critical components in the matching algorithm of Edmonds [Canad. J. Math., 17 (1965), pp. 449–467]. The running time of the algorithm is $\mathrm{O}(n^4 Q)$, where $n$ is the number of vertices and $Q$ is the time for an independence test. The algorithm also gives a common generalization of the Edmonds–Gallai decomposition for matchings and the principal partition for matroid intersection.

Multidimensional Manhattan Street Networks

F. Comellas, C. Dalfó, and M. A. Fiol

SIAM J. Discrete Math. 22, pp. 1428-1447 (20 pages)

Online Publication Date: September 19, 2008

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We formally define the $n$-dimensional Manhattan street network $M_n$—a special case of an $n$-regular digraph—and we study some of its structural properties. In particular, we show that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dimensional version of the wallpaper group $pgg$. These results induce a useful new representation of $M_n$, which can be applied to design a local (shortest-path) routing algorithm and to study some other metric properties, such as the diameter. We also show that the $n$-dimensional Manhattan street networks are Hamiltonian and, in the standard case (that is, in dimension two), we give sufficient conditions for a $2$-dimensional Manhattan street network to be decomposable into two arc-disjoint Hamiltonian cycles.

Partitions of Faulty Hypercubes into Paths with Prescribed Endvertices

Tomáš Dvořák and Petr Gregor

SIAM J. Discrete Math. 22, pp. 1448-1461 (14 pages)

Online Publication Date: September 25, 2008

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Given a set $\pc=\{a_i,b_i\}_{i=1}^m$ of pairs of vertices in a graph $G$, is there a collection of paths $\{P_i\}_{i=1}^m$ such that $P_i$ connects $a_i$ with $b_i$ and $\{V(P_i)\}_{i=1}^m$ partitions $V(G)$? We study this problem for the graph $Q_n-\ff$ obtained from the $n$-dimensional hypercube $Q_n$ by removing a set $\ff$ of faulty vertices. We show that an obvious necessary condition for the existence of such a partition is also sufficient provided $2|\pc|+ 3|\ff|\le n-3$. As a corollary, we obtain a similar characterization for the existence of a hamiltonian cycle and a hamiltonian path of $Q_n-\ff$ provided $|\ff|\le(n-5)/3$. On the other hand, if the size of $\ff$ is not limited, the problems are NP-complete.

Total-Coloring of Plane Graphs with Maximum Degree Nine

Łukasz Kowalik, Jean-Sébastien Sereni, and Riste Škrekovski

SIAM J. Discrete Math. 22, pp. 1462-1479 (18 pages)

Online Publication Date: October 01, 2008

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The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree $\Delta$ admits a $(\Delta+2)$-total-coloring. Similar to edge-colorings—with Vizing's edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if $\Delta\ge10$, then every plane graph of maximum degree $\Delta$ is $(\Delta+1)$-totally-colorable. On the other hand, such a statement does not hold if $\Delta\le3$. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.

Odd Minimum Cut Sets and $b$-Matchings Revisited

Adam N. Letchford, Gerhard Reinelt, and Dirk Oliver Theis

SIAM J. Discrete Math. 22, pp. 1480-1487 (8 pages) | Cited 1 time

Online Publication Date: October 01, 2008

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The famous Padberg–Rao separation algorithm for $b$-matching polyhedra can be implemented to run in $\mathcal{O}(|V|^2|E|\log(|V|^2/|E|))$ time in the uncapacitated case, and in $\mathcal{O}(|V||E|^2\log(|V|^2/|E|))$ time in the capacitated case. We give a new and simple algorithm for the capacitated case which can be implemented to run in $\mathcal{O}(|V|^2|E|\log(|V|^2/|E|))$ time.

Analysis of Serial Turbo Codes over Abelian Groups for Symmetric Channels

Federica Garin and Fabio Fagnani

SIAM J. Discrete Math. 22, pp. 1488-1526 (39 pages)

Online Publication Date: October 01, 2008

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In this paper we study serial turbo interconnections of Abelian group codes to be used on symmetric channels. Particular attention is devoted to AWGN channel with input restricted to $m$-PSK constellation with corresponding group structure $\mathbb{Z}_m$. We establish the exact asymptotic decay of the average symbol and word error probabilities when the interleaver length goes to infinity (interleaver gain). Moreover, we give a detailed characterization of the distance parameter characterizing the behavior for the signal-to-noise ratio going to infinity (effective free distance). Some of our results are new also in the binary context; in particular, the lower bound to the error probability decay.

Tropical Linear Spaces

David E. Speyer

SIAM J. Discrete Math. 22, pp. 1527-1558 (32 pages) | Cited 1 time

Online Publication Date: October 17, 2008

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We define the tropical analogues of the notions of linear spaces and Plücker coordinates 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 is 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 conjecture.

Bounds for the Real Number Graph Labellings and Application to Labellings of the Triangular Lattice

Daniel Král' and Petr Škoda

SIAM J. Discrete Math. 22, pp. 1559-1569 (11 pages) | Cited 2 times

Online Publication Date: October 17, 2008

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We establish new lower and upper bounds for the real number graph labelling problem. As an application, we consider a problem of Griggs to determine the optimum spans of $L(p,q)$-labellings of the infinite triangular plane lattice and find (using a computer) the optimum spans for all $p$ and $q$.

Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends

Xiao Zhou and Takao Nishizeki

SIAM J. Discrete Math. 22, pp. 1570-1604 (35 pages)

Online Publication Date: October 17, 2008

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In an orthogonal drawing of a planar graph $G$, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of $G$ is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of $G$. In this paper we give an algorithm to find an optimal orthogonal drawing of any given series-parallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.

How Many Points Can Be Reconstructed from $k$ Projections?

Jiří Matoušek, Aleš Přívětivý, and Petr Škovroň

SIAM J. Discrete Math. 22, pp. 1605-1623 (19 pages)

Online Publication Date: October 17, 2008

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Let $A$ be an $n$-point set in the plane. A discrete X-ray of $A$ in direction $u$ gives the number of points of $A$ on each line parallel to $u$. We define $F(k)$ as the maximum number $n$ such that there exist $k$ directions $u_1,\dots,u_k$ such that every set of at most $n$ points in the plane can be uniquely reconstructed from its discrete X-rays in these directions. A simple “cube” construction shows $F(k)\le2^{k-1}$. We establish the lower bound $F(k)\ge2^{\Omega(k/\log k)}$ by reducing the problem through linear algebra to a graph-theoretic question, for which we then obtain an almost tight bound. As a part of the proof we establish a result in extremal theory that allows one to conclude that, under certain conditions, a graph has only at most a logarithmic density, which may be of independent interest. We also improve the upper bound to $F(k)\le O(1.81712^k)$ (or $O(1.79964^k)$ if we allow $A$ to be a multiset).

Minimum Cost Homomorphisms to Semicomplete Bipartite Digraphs

Gregory Gutin, Arash Rafiey, and Anders Yeo

SIAM J. Discrete Math. 22, pp. 1624-1639 (16 pages)

Online Publication Date: October 17, 2008

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For digraphs $D$ and $H$, a mapping $f:V(D)\rightarrow V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H)$. If, moreover, each vertex $u\in V(D)$ is associated with costs $c_i(u)$, $i\in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the minimum cost homomorphism problem for $H$. The problem is to decide, for an input graph $D$ with costs $c_i(u)$, $u\in V(D)$, $i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete bipartite digraphs $H$. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a $k$-Min-Max ordering of digraphs.

On the Polytope of the (1,2)-Survivable Network Design Problem

Mohamed Didi Biha, Hervé Kerivin, and A. Ridha Mahjoub

SIAM J. Discrete Math. 22, pp. 1640-1666 (27 pages)

Online Publication Date: October 17, 2008

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This paper deals with the survivable network design problem where each node $v$ has a connectivity type $r(v)$ equal to 1 or 2, and the survivability conditions require the existence of at least $\min\{r(s),r(t)\}$ edge-disjoint paths for all distinct nodes $s$ and $t$. We consider the polytope given by the trivial and cut inequalities together with the partition inequalities. More precisely, we study some structural properties of this polytope which leads us to give some sufficient conditions for this polytope to be integer in the class of series-parallel graphs. With both separation problems for the cut and partition inequalities being polynomially solvable, we then obtain a polynomial time algorithm for the (1,2)-survivable network design problem in a subclass of series-parallel graphs including the outerplanar graph class. We also introduce a new class of facet-defining inequalities for the polytope associated to the (1,2)-survivable network design problem.
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