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Interval graphs admit elegant structural characterizations and linear time recognition algorithms; on the other hand, the usual interval digraphs lack a forbidden structure characterization as well as a low-degree polynomial time... more
Interval graphs admit elegant structural characterizations and linear time recognition algorithms; on the other hand, the usual interval digraphs lack a forbidden structure characterization as well as a low-degree polynomial time recognition algorithm. In this paper we identify another natural digraph analogue of interval graphs that we call ”chronological interval digraphs”. By contrast, the new class admits both a forbidden structure characterization and a linear time recognition algorithm. Chronological interval digraphs arise by interpreting the standard definition of an interval graph with a natural orientation of its edges. Specifically, $G$ is a chronological interval digraph if there exists a family of closed intervals $I_v$, $v \in V(G)$, such that $uv$ is an arc of $G$ if and only if $I_u$ intersects $I_v$ and the left endpoint of $I_u$ is not greater than the left endpoint of $I_v$. (Equivalently, if and only if $I_u$ contains the left endpoint of $I_v$.)We characterize c...
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Given any set $\mathcal{B}$ of complete bipartite graphs, we ask whether a graph H admits a $\mathcal{B}$-factor, i.e., a spanning subgraph, each of whose components is a member of $\mathcal{B}$. More generally, we seek in H a maximum... more
Given any set $\mathcal{B}$ of complete bipartite graphs, we ask whether a graph H admits a $\mathcal{B}$-factor, i.e., a spanning subgraph, each of whose components is a member of $\mathcal{B}$. More generally, we seek in H a maximum $\mathcal{B}$-packing, i.e., a $\mathcal{B}$-factor of a maximum size subgraph of H. We first treat the interesting special case when $\mathcal{B}$ is a set of stars. The results are generalized to arbitrary $\mathcal{B}$ in the last section. We prove for most of these problems that they are $\mathcal{N}\mathcal{P}$-hard; we also show that the remaining problems admit polynomial algorithms based on augmenting configurations. The simplicity of these algorithms, as well as the implied min-max theorems, resemble the theory of matchings in bipartite, rather than general, graphs.
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Graph homomorphisms have been investigated since the nineteen sixties [l, 2, 4, 7–10, 12–14, 16–20], and have enjoyed renewed interest lately, because of their relation to grammar interpretations [16]. (A different notion of graph... more
Graph homomorphisms have been investigated since the nineteen sixties [l, 2, 4, 7–10, 12–14, 16–20], and have enjoyed renewed interest lately, because of their relation to grammar interpretations [16]. (A different notion of graph homomorphism was investigated by Dirac, Wagner, and others, [3, 21].) A homomorphism G → H is a mapping of the vertex set of G to the vertex set of H such that adjacent vertices have adjacent images. Because a homomorphism c: G → K n is just an n-coloring of G , a homomorphism G + H is also called an H-coloring of G. The following H-coloring problem has been the object of recent interest: Instance : A graph G. Question : Is it possible to H -color the graph G ? Several authors have studied the complexity of the H -coloring problem for various (families of) fixed graphs H [l, 2, 17, 18]. Since there is an easy H -colorability test when H is bipartite, and since all other examples of the H -colorability problem that were treated (complete graphs, odd cycles, complements of odd cycles, etc., [l, 17, 18]) turned out to be NP -complete, the natural conjecture, formulated in several sources [17, 18] (including David Johnson's NP -completeness column [12]) asserts that the H -coloring problem is NP -complete for any non-bipartite graph H. We have proved this conjecture, and will publish a full proof elsewhere [ll]. Here, we outline the ideas behind our proof.
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Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T so that no two paths of the same... more
Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T so that no two paths of the same colour use the same arc of T is equal to the maximum number of paths passing through an arc of T. The proof implies a polynomial algorithm for actually assigning the minimum number of colours. When only a subset of the directed paths is to be coloured, the problem is known to be NP-complete; we describe certain instances of the problem which can be eeciently solved. These results are applied to WDM (wavelength-division multiplexing) routing in all-optical networks. In particular, we solve the all-to-all gossiping problem in optical networks.
The need for sorting algorithms which operate in a fixed number of rounds (rather than have each new comparison depend on the outcomes of all previous comparisons) arises in structural modeling. Since all comparisons within a round are... more
The need for sorting algorithms which operate in a fixed number of rounds (rather than have each new comparison depend on the outcomes of all previous comparisons) arises in structural modeling. Since all comparisons within a round are evaluated simultaneously, such algorithms have an obvious connection to parallel processing.In an earlier paper (SIAM J. Comput.,10 (1981), pp. 465–472) we used a counting argument to prove the existence of subquadratic sorting algorithms for two rounds. Here we develop optimal algorithms for merging in rounds, and apply them to actually construct good sorting algorithms for k rounds, $k\geqq 3$. For example, in $k = 66$ rounds, our algorithm will sort any n-element linearly ordered set with $O ( n^{1.10} )$ comparisons.
The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A.... more
The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). Egri et al. (SODA 2014) augmented this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in log space or is hard for NL. A conjecture of Larose and Tesson from 2007 forecasts that when LHOM(H) is in log space, then in fact, it falls in a small subclass of log space, the set of problems expressible in symmetric Data log. The present work verifies the conjecture for LHOM(H) (and, indeed, for the wider class of conservative CSPs with binary constraints), and by so doing sharpens the aforementioned dichotomy. A combinatorial characterization of symmetric Data log provides the language in which the algorithmic ideas of the paper, quite different from the ones in Egri et al., are formalized.
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Graph colourings may be viewed as special constraint satisfaction problems. The class of k-colouring problems enjoys a well known dichotomy of complexitythese problems are polynomial time solvable when k≤ 2, and NP-complete when k≥ 3.... more
Graph colourings may be viewed as special constraint satisfaction problems. The class of k-colouring problems enjoys a well known dichotomy of complexitythese problems are polynomial time solvable when k≤ 2, and NP-complete when k≥ 3. For general ...
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We prove that there exist graphs with n vertices and at most $2n^{5/3} \log n$ edges for which every acyclic orientation has in its transitive closure at least $\begin{pmatrix} n \\ 2 \end{pmatrix} - 10n^{5/3} $ arcs. We conclude that... more
We prove that there exist graphs with n vertices and at most $2n^{5/3} \log n$ edges for which every acyclic orientation has in its transitive closure at least $\begin{pmatrix} n \\ 2 \end{pmatrix} - 10n^{5/3} $ arcs. We conclude that with $2n^{5/3} \log n$ parallel processors n items may be sorted with all comparisons arranged in two time intervals. We also show that $\frac{1}{9}n^{3/2} $ processors are not sufficient to achieve the same end. These results are extended to parallel sorting in k time intervals, and related to other work on parallel sorting. The existence of sorting algorithms achieving the bounds is proved by nonconstructive methods. (The constants quoted in the abstract are somewhat improved in the paper.)
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Annals of Discrete Mathematics 12 (1982) 155-168 © North-Holland Publishing Company ANALOGUES OF THE SHANNON CAPACITY OF A GRAPH Pavol HELL* and Fred S. ROBERTS** Rutgers Vniversity. New Brunswick, NJ 08903, USA Dedicated to Professor A.... more
Annals of Discrete Mathematics 12 (1982) 155-168 © North-Holland Publishing Company ANALOGUES OF THE SHANNON CAPACITY OF A GRAPH Pavol HELL* and Fred S. ROBERTS** Rutgers Vniversity. New Brunswick, NJ 08903, USA Dedicated to Professor A. Kotzig on the ...
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ABSTRACT Broadcasting is an information dissemination process in which a message originating at one node of a communication network (modeled as an undirected graph) is sent to all other nodes by means of calls involving two nodes at a... more
ABSTRACT Broadcasting is an information dissemination process in which a message originating at one node of a communication network (modeled as an undirected graph) is sent to all other nodes by means of calls involving two nodes at a time, with each node participating in at most one call at any time. We are interested in efficient broadcasting in a class of cubic graphs known as generalized chordal rings. These graphs have been found useful for having a small diameter D, among graphs with a given number of vertices and maximum degree. We show that the minimum broadcast time in any generalized chordal ring is D, D + 1, or D + 2. For the generalized chordal rings of diameter D which have the greatest (or greatest-known) number of nodes, we then evaluate exactly the minimum broadcast time. It turns out to be D + 1 when D is even and D + 2 when D is odd. For these purposes, we review the construction of these extremal generalized chodal rings. We also review the optimal broadcast schemes for infinite triangular grids, which we use to prove our bounds. Finally, we ask for the maximum number of nodes that can be informed by a broadcast in time t in any generalized chordal ring. We answer this completely for even t and almost completely for odd t. We use a geometric approach, based on plane tessellations. © 2003 Wiley Periodicals, Inc.
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ABSTRACT
We discuss connections between sorting algorithms and graph theory. Taking a graph-theoretic view of sorting, we motivate generalizations of the problem of serial sorting, and describe recent results on parallel sorting, and sorting in... more
We discuss connections between sorting algorithms and graph theory. Taking a graph-theoretic view of sorting, we motivate generalizations of the problem of serial sorting, and describe recent results on parallel sorting, and sorting in rounds, which depend on deep graph-theoretic results.