Parameterized Problems Related to Seidel's Switching

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

Download Full-text

Classes of graphs with restricted interval models

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.263 ◽

1999 ◽

Vol Vol. 3 no. 4 ◽

Author(s):

Andrzej Proskurowski ◽

Jan Arne Telle

Keyword(s):

Maximum Clique ◽

Interval Graphs ◽

Induced Subgraph ◽

Graph Theoretic ◽

Graph Classes ◽

International Audience ◽

Forbidden Induced Subgraph ◽

Nested Hierarchy ◽

Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

Download Full-text

Long cycles in hypercubes with distant faulty vertices

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.466 ◽

2009 ◽

Vol Vol. 11 no. 1 (Graph and Algorithms) ◽

Author(s):

Petr Gregor ◽

Riste Škrekovski

Keyword(s):

Linear Code ◽

Minimum Distance ◽

Hamiltonian Cycle ◽

Minimum Degree ◽

Induced Subgraphs ◽

Induced Subgraph ◽

International Audience ◽

Long Cycles

Graphs and Algorithms International audience In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.

Download Full-text

Popular Matchings in Complete Graphs

Algorithmica ◽

10.1007/s00453-020-00791-7 ◽

2021 ◽

Author(s):

Ágnes Cseh ◽

Telikepalli Kavitha

Keyword(s):

Complete Graph ◽

The Other ◽

Complete Graphs ◽

Matching Problem ◽

Graph Theoretic ◽

The Given ◽

Popular Matching

AbstractOur input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching $$M'$$ M ′ : here each vertex casts a vote for the matching in $$\{M,M'\}$$ { M , M ′ } in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes $$\texttt {NP}$$ NP -complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and $$\texttt {NP}$$ NP -complete when n has the other parity.

Download Full-text

Balanced Avoidance Games on Random Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3454 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Martin Marciniszyn ◽

Dieter Mitsche ◽

Miloš Stojaković

Keyword(s):

Random Graphs ◽

Complete Graph ◽

Optimal Strategy ◽

Random Order ◽

The Other ◽

Threshold Function ◽

Graph Theoretic ◽

International Audience ◽

Balanced Coloring ◽

Monochromatic Copy

International audience We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with $n$ vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vice versa. His goal is to avoid a monochromatic copy of a given fixed graph $H$ in both colors for as long as possible. The game ends as soon as the first monochromatic copy of $H$ has appeared. We show that the duration of the game is determined by a threshold function $m_H = m_H(n)$. More precisely, Painter will asymptotically almost surely (a.a.s.) lose the game after $m = \omega (m_H)$ edge pairs in the process. On the other hand, there is an essentially optimal strategy, that is, if the game lasts for $m = o(m_H)$ moves, then Painter will a.a.s. successfully avoid monochromatic copies of H using this strategy. Our attempt is to determine the threshold function for certain graph-theoretic structures, e.g., cycles.

Download Full-text

Krausz dimension and its generalizations in special graph classes

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.623 ◽

2013 ◽

Vol Vol. 15 no. 1 (Graph Theory) ◽

Author(s):

Olga Glebova ◽

Yury Metelsky ◽

Pavel Skums

Keyword(s):

Open Problem ◽

Chordal Graphs ◽

Np Hard ◽

Induced Subgraphs ◽

Fixed Parameter Tractable ◽

Graph Classes ◽

Fixed Parameter ◽

Minimum Number ◽

International Audience ◽

Split Graphs

Graph Theory International audience A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

Download Full-text

NP-Completeness Results for Minimum Planar Spanners

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.257 ◽

1998 ◽

Vol Vol. 3 no. 1 ◽

Author(s):

Ulrik Brandes ◽

Dagmar Handke

Keyword(s):

Integrative statistical analyses of multiple liquid biopsy analytes in metastatic breast cancer

Genome Medicine ◽

10.1186/s13073-021-00902-1 ◽

2021 ◽

Vol 13 (1) ◽

Author(s):

Corinna Keup ◽

Vinay Suryaprakash ◽

Siegfried Hauch ◽

Markus Storbeck ◽

Peter Hahn ◽

...

Keyword(s):

Breast Cancer ◽

Overall Survival ◽

Metastatic Breast Cancer ◽

Mutual Information ◽

Hierarchical Clustering ◽

Liquid Biopsy ◽

Metastatic Breast ◽

The Other ◽

Theoretic Analysis ◽

Graph Theoretic

Abstract Background Single liquid biopsy analytes (LBAs) have been utilized for therapy selection in metastatic breast cancer (MBC). We performed integrative statistical analyses to examine the clinical relevance of using multiple LBAs: matched circulating tumor cell (CTC) mRNA, CTC genomic DNA (gDNA), extracellular vesicle (EV) mRNA, and cell-free DNA (cfDNA). Methods Blood was drawn from 26 hormone receptor-positive, HER2-negative MBC patients. CTC mRNA and EV mRNA were analyzed using a multi-marker qPCR. Plasma from CTC-depleted blood was utilized for cfDNA isolation. gDNA from CTCs was isolated from mRNA-depleted CTC lysates. CTC gDNA and cfDNA were analyzed by targeted sequencing. Hierarchical clustering was performed within each analyte, and its results were combined into a score termed Evaluation of multiple Liquid biopsy analytes In Metastatic breast cancer patients All from one blood sample (ELIMA.score), which calculates the contribution of each analyte to the overall survival prediction. Singular value decomposition (SVD), mutual information calculation, k-means clustering, and graph-theoretic analysis were conducted to elucidate the dependence between individual analytes. Results A combination of two/three/four LBAs increased the prevalence of patients with actionable signals. Aggregating the results of hierarchical clustering of individual LBAs into the ELIMA.score resulted in a highly significant correlation with overall survival, thereby bolstering evidence for the additive value of using multiple LBAs. Computation of mutual information indicated that none of the LBAs is independent of the others, but the ability of a single LBA to describe the others is rather limited—only CTC gDNA could partially describe the other three LBAs. SVD revealed that the strongest singular vectors originate from all four LBAs, but a majority originated from CTC gDNA. After k-means clustering of patients based on parameters of all four LBAs, the graph-theoretic analysis revealed CTC ERBB2 variants only in patients belonging to one particular cluster. Conclusions The additional benefits of using all four LBAs were objectively demonstrated in this pilot study, which also indicated a relative dominance of CTC gDNA over the other LBAs. Consequently, a multi-parametric liquid biopsy approach deconvolutes the genomic and transcriptomic complexity and should be considered in clinical practice.

Download Full-text

Scalable mining of maximal quasi-cliques

Proceedings of the VLDB Endowment ◽

10.14778/3436905.3436916 ◽

2020 ◽

Vol 14 (4) ◽

pp. 573-585

Author(s):

Guimu Guo ◽

Da Yan ◽

M. Tamer Özsu ◽

Zhe Jiang ◽

Jalal Khalil

Keyword(s):

Graph Mining ◽

State Of The Art ◽

Minimum Degree ◽

The Other ◽

Dense Subgraph ◽

Load Imbalance ◽

Subgraph Mining ◽

Execution Engine ◽

Clique Algorithm ◽

Dense Subgraph Mining

Given a user-specified minimum degree threshold γ , a γ -quasiclique is a subgraph g = (V g , E g ) where each vertex ν ∈ V g connects to at least γ fraction of the other vertices (i.e., ⌈ γ · (| V g |- 1)⌉ vertices) in g. Quasi-clique is one of the most natural definitions for dense structures useful in finding communities in social networks and discovering significant biomolecule structures and pathways. However, mining maximal quasi-cliques is notoriously expensive. In this paper, we design parallel algorithms for mining maximal quasi-cliques on G-thinker, a distributed graph mining framework that decomposes mining into compute-intensive tasks to fully utilize CPU cores. We found that directly using G-thinker results in the straggler problem due to (i) the drastic load imbalance among different tasks and (ii) the difficulty of predicting the task running time. We address these challenges by redesigning G-thinker's execution engine to prioritize long-running tasks for execution, and by utilizing a novel timeout strategy to effectively decompose long-running tasks to improve load balancing. While this system redesign applies to many other expensive dense subgraph mining problems, this paper verifies the idea by adapting the state-of-the-art quasi-clique algorithm, Quick, to our redesigned G-thinker. Extensive experiments verify that our new solution scales well with the number of CPU cores, achieving 201× runtime speedup when mining a graph with 3.77M vertices and 16.5M edges in a 16-node cluster.

Download Full-text