The Vertex Separation and Search Number of a Graph

Ellis, John; Sudborough, Ivan Hal; Turner, Jonathan S.

doi:10.1006/inco.1994.1064

Cited by 195 publications

(150 citation statements)

References 18 publications

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“…The linear-time algorithm for computing the path-width of trees (and hence their linear rank-width by [1]) is based on the following characterization. Proposition 3.1 (Ellis, Sudborough, and Turner [13]). A tree T has path-width at most k if and only if for every vertex v of T at most two components of T zv have path-width at most k, and all the other components have path-width at most k´1.…”

Section: Limbs In Canonical Decompositionsmentioning

confidence: 99%

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Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

2016

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Abstract. Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank-width at most 1, can be computed in time Opn 2¨l og 2 nq, and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every n-element matroid of branch-width at most 2 can be computed in time Opn 2¨l og 2 nq, provided that the matroid is given by its binary representation.To establish this result, we present a characterization of the linear rankwidth of distance-hereditary graphs in terms of their canonical split decompositions. This characterization is similar to the known characterization of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex separation and search number of a graph. Inf. Comput., 113(1): 1994]. However, different from forests, it is non-trivial to relate substructures of the canonical split decomposition of a graph with some substructures of the given graph. We introduce a notion of 'limbs' of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization.

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Section: Limbs In Canonical Decompositionsmentioning

confidence: 99%

“…While a similar idea has been exploited in [1,13,26] for other parameters, here we encounter a new problem. When we take a subgraph of a given split decomposition, the obtained split decomposition may have vertices that do not represent vertices of the original graph.…”

Section: Introductionmentioning

confidence: 99%

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

2016

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“…According to [34], if the problem of checking whether p(G) ≤ k is NP-complete, then |O ≤m p,k | is a super-polynomial function of k, unless the polynomial hierarchy collapses to Σ P 3 . Characterizations of p(G) ≤ k (yielding better lower bounds for |O ≤m p,k |) have been provided for several parameters [10,20,40,58,84,98,99,115,117,118]. However, to our knowledge, there is not yet a natural parameter p for which a complete characterization of O ≤m p,k is known.…”

Section: Minor-closed Graph Parametersmentioning

confidence: 99%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

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“…Originally, the decontamination problem has been introduced in [7,17] and has been extensively studied in the literature under the term graph search (e.g., see [10,[14][15][16]18]). The difference with the CMD problem is that searchers do not necessarily move in a connected way, i.e., may 'jump' from one node to another.…”

Section: Upgrading Vs Decontaminatingmentioning

confidence: 99%

Secure Upgrade of Hardware Security Modules in Bank Networks

Focardi

Luccio

2010

Automated Reasoning for Security Protocol Analysis and Issues in the Theory of Security

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Abstract. We study the secure upgrade of critical components in wide networked systems, focussing on the case study of PIN processing Hardware Security Modules (HSMs). These tamper-resistant devices, used by banks to securely transmit and verify the PIN typed at the ATMs, have been shown to suffer from API level attacks that allow an insider to recover user PINs and, consequently, clone cards. Proposed fixes require to reduce and modify the HSM functionality by, e.g., sticking on a single format of the transmitted PIN or adding MACs for the integrity of user data. Upgrading HSMs worldwide is, of course, unaffordable. We thus propose strategies to incrementally upgrade the network so to obtain upgraded, secure subnets, while preserving the compatibility towards the legacy system. Our strategies aim at finding tradeoffs between the cost for special "guardian" HSMs used on the borderline between secure and insecure nodes, and the size of the team working in the upgrade process, representing the maximum number of nodes that can be simultaneously upgraded.

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The Vertex Separation and Search Number of a Graph

Cited by 195 publications

References 18 publications

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

Graph Minors and Parameterized Algorithm Design

Secure Upgrade of Hardware Security Modules in Bank Networks

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