Dual Connectedness of Edge-Bicolored Graphs and Beyond

Cai, Lilong; Ye, Junjie

doi:10.1007/978-3-662-44465-8_13

Cited by 12 publications

(28 citation statements)

References 20 publications

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“…Notice that this notion is different from the usual notion of bicolored graph ( [17]) in that a single edge is allowed to take both "colors" at the same time, namely an edge (i, j) may belong both to E o and E p . This is needed in order to accommodate for standard bilateral interactions within this generalized framework.…”

Section: Graph Theoretical Preliminariesmentioning

confidence: 99%

Consensus for nonlinear monotone networks with unilateral interactions

Manfredi

Angeli

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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Abstract-This paper deals with an extended framework of the distributed asymptotic agreement problem by allowing the presence of unilateral interactions (optimistic or pessimistic) in place of bilateral ones, for a large class of nonlinear monotone time-varying networks. In this original setup we firstly introduce notions of unilateral optimistic and/or pessimistic interaction, of associated bicolored edge in the interaction graph and a suitable graph-theoretical connectedness property. Secondly, we formulate a new assumption of integral connectivity and show that it is sufficient to guarantee exponential convergence towards the agreement subspace. Finally, we remark that the proposed conditions are also necessary for consensuability. Theoretical advances are emphasized through illustrative examples given both to support the discussion and to highlight how the proposed framework extends all existing conditions for consensus of monotone networks.

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Section: Graph Theoretical Preliminariesmentioning

confidence: 99%

Consensus for nonlinear monotone networks with unilateral interactions

Manfredi

Angeli

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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“…In this connection, Raman and Sikdar [17] have studied the parameterized complexity of Induced Π -Subgraph on digraphs, and Cai and Ye [4] have recently shown the W[1]-hardness of determining whether a digraph contains a strongly connected induced subgraph on k vertices. Problem 1.…”

Section: Conjecture 1 Connected Induced π -Subgraph Is W[1]-hard Formentioning

confidence: 99%

Parameterized complexity of finding connected induced subgraphs

Cai

2015

Theoretical Computer Science

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For a graph property Π , i.e., a family Π of graphs, the Connected Induced Π -Subgraph problem asks whether an input graph G contains k vertices V such that the induced subgraph G[V ] is connected and satisfies property Π .In this paper, we study the parameterized complexity of Connected Induced Π -Subgraph for decidable hereditary properties Π , and give a nearly complete characterization in terms of whether Π includes all complete graphs, all stars, and all paths. As a consequence, we obtain a complete characterization of the parameterized complexity of our problem when Π is the family of H-free graphs for a fixed graph H with h ≥ 3 vertices: W[1]-hard ifH is K h , K h , or K 1,h−1 ; and FPT otherwise. Furthermore, we also settle the parameterized complexity of the problem for many well-known families Π of graphs: FPT for perfect graphs, chordal graphs, and interval graphs, but W[1]-hard for forests, bipartite graphs, planar graphs, line graphs, and degree-bounded graphs.

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“…For a family of graphs F, given a graph G and an integer k, the F-deletion (Edge F-deletion) problem asks whether we can delete at most k vertices (edges) in G so that the resulting graph belongs to F. The F-deletion (Edge F-deletion) problems generalize many of the NP-hard problems like Vertex Cover, Feedback vertex set, Odd cycle transversal, Edge Bipartization, etc. Inspired by applications, Cai and Ye introduced variants of F-deletion (Edge Fdeletion) problems on edge colored graph [7]. Edge colored graphs are studied in graph theory with respect to various problems like Monochromatic and Heterochromatic Subgraphs [15], Alternating paths [6,8,20], Homomorphism in edge-colored graphs [3], Graph Partitioning in 2-edge colored graphs [5] etc.…”

Section: Introductionmentioning

confidence: 99%

“…Cai and Ye studied the Dually Connected Induced subgraph and Dual Separator on 2-edge colored graphs [7]. Agrawal et al [1] studied a simultaneous variant of Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set, in the realm of parameterized complexity.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Feedback Edge Set: A Parameterized Perspective

et al. 2020

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In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G) → 2 [α] , and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i ∈ [α], G i − S is acyclic. Here, G i = (V (G), {e ∈ E(G) | i ∈ col(e)}) and [α] = {1,. .. , α}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i ∈ [α], G i − S is acyclic. Unlike the vertex variant of the problem, when α = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for α = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2 o(k) n O(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2 ωkα+α log k n O(1)), where ω is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when α = 2. We also give a kernel for Sim-FES with (kα) O(α) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G) → 2 [α]. The question is whether there is a edge subset F of cardinality at least q in G such that for all i ∈ [α], G[F i ] is acyclic. Here, F i = {e ∈ F | i ∈ col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2 ωqα n O(1)). All our algorithms are based on parameterized version of the Matroid Parity problem.

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Dual Connectedness of Edge-Bicolored Graphs and Beyond

Cited by 12 publications

References 20 publications

Consensus for nonlinear monotone networks with unilateral interactions

Consensus for nonlinear monotone networks with unilateral interactions

Parameterized complexity of finding connected induced subgraphs

Simultaneous Feedback Edge Set: A Parameterized Perspective

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