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### Solving Graph Isomorphism Using Parameterized Matching [chapter]

Juan Mendivelso, Sunghwan Kim, Sameh Elnikety, Yuxiong He, Seung-won Hwang, Yoan Pinzón
2013 Lecture Notes in Computer Science
We propose a new approach to solve graph isomorphism using parameterized matching.  ...  To find isomorphism between two graphs, one graph is linearized, i.e., represented as a graph walk that covers all nodes and edges such that each element is represented by a parameter.  ...  A good algorithm for graph isomorphism should find isomorphic graphs quickly in many cases. In this paper we apply parameterized matching to solve graph isomorphism.  ...

### On the complexity of various parameterizations of common induced subgraph isomorphism

Faisal N. Abu-Khzam, Édouard Bonnet, Florian Sikora
2017 Theoretical Computer Science
feedback vertex sets of the input graphs, that is solving them in polynomial time is very unlikely when this parameter is a constant.  ...  MCIS remains NP-hard on many graph classes including bipartite graphs, planar graphs and k-trees. Little is known, however, about the parameterized complexity of the problem.  ...  However, that algorithm cannot help us much for solving MCCIS since it relies on the existence of a feasible solution of size at least ≈ n − k which consists of mapping the two big independent sets of  ...

### Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)

Dániel Marx, Michal Pilipczuk, Marc Herbstritt
2014 Symposium on Theoretical Aspects of Computer Science
Given two graphs H and G, the Subgraph Isomorphism problem asks if H is isomorphic to a subgraph of G. While NP-hard in general, algorithms exist for various parameterized versions of the problem.  ...  Our goal is to systematically investigate the possible parameterized algorithms that can exist for Subgraph Isomorphism.  ...  In fact, the only way we were able to solve these matching problems is by the randomized algebraic matching algorithm of Mulmuley et al. [28] .  ...

### On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism [chapter]

Faisal N. Abu-Khzam, Édouard Bonnet, Florian Sikora
2015 Lecture Notes in Computer Science
It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS.  ...  We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2 o(k log k) .  ...  Obviously, if there is a k-clique S in G, then the graph I(G)[S ∪ E(S)] is isomorphic to I(K k ). Now, let us assume that I(K k ) is isomorphic to an induced subgraph of I(G).  ...

### On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism [article]

Faisal N. Abu-Khzam, Édouard Bonnet, Florian Sikora
2017 arXiv   pre-print
It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS.  ...  We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2^o(k k).  ...  Obviously, if there is a k-clique S in G, then the graph I(G)[S ∪ E(S)] is isomorphic to I(K k ). Now, let us assume that I(K k ) is isomorphic to an induced subgraph of I(G).  ...

### Foreword: Special Issue on IPEC 2014

Marek Cygan, Pinar Heggernes
2016 Algorithmica
In The Parameterized Complexity of Geometric Graph Isomorphism, V.  ...  Jannis Bulian and Anuj Dawar consider the Graph Isomorphism problem in Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree.  ...

### Parameterized Aspects of Strong Subgraph Closure

Petr A. Golovach, Pinar Heggernes, Athanasios L. Konstantinidis, Paloma T. Lima, Charis Papadopoulos
2020 Algorithmica
Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study  ...  An induced matching, denoted byqK 2 , is a matching M of q edges such that G[V (M )] is isomorphic to qK 2 .Let us give a couple of observations on the nature of our problem.An F -graph of a subgraph H  ...  If two graphs H and F are isomorphic then we write H F , and if they are not isomorphic then we write H F .  ...

### Optimality and tight results in parameterized complexity (Dagstuhl Seminar 14451)

Stefan Kratsch, Daniel Lokshtanov, Dániel Marx, Peter Rossmanith, Marc Herbstritt
2015 Dagstuhl Reports
Over the last two decades parameterized complexity has become one of the main tools for handling intractable problems.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 14451 "Optimality and tight results in parameterized complexity".  ...  For both of these families, ETH can be used to pinpoint the optimum form of the running time of a parameterized algorithm solving the problem.  ...

### Isomorphism for Graphs of Bounded Feedback Vertex Set Number [chapter]

Stefan Kratsch, Pascal Schweitzer
2010 Lecture Notes in Computer Science
We prove that the Graph Isomorphism problem is fixed-parameter tractable when parameterized by the feedback vertex set number.  ...  For graphs of bounded feedback vertex set number the presented algorithm runs in time O(n 2 ), and is thereby the first fixed-parameter tractable Graph Isomorphism algorithm for a non-trivial parameter  ...  We now design an FPT-algorithm that solves Graph Isomorphism parameterized with the feedback vertex set number.  ...

### Stable Matching Games: Manipulation via Subgraph Isomorphism

Sushmita Gupta, Sanjukta Roy
2017 Algorithmica
Our algorithm is a nontrivial combination of a parameterized algorithm for Subgraph Isomorphism, a relationship between stable matching and finding an out-branching in an appropriate graph and enumerating  ...  We will use (L M , L W ) to denote the set of preference lists of men and women, and the men-optimal matching with respect to these lists is denoted by GS(L M , L W ).  ...  graph [2, 17] , and a parameterized algorithm for Colored Subgraph Isomorphism [1, 7, 8] , we can search for a subgraph in the universal suitor graph that is isomorphic to an out-branching corresponding  ...

### Graph Square Roots of Small Distance from Degree One Graphs [article]

Petr A. Golovach, Paloma T. Lima, Charis Papadopoulos
2020 arXiv   pre-print
We are interested in the parameterized complexity of the problem for classes ℋ that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is,  ...  In particular, we prove that the VC-k Root problem, that asks whether an input graph has a square root with vertex cover of size at most k, cannot be solved in time 2^2^o(k)· n^O(1) unless Exponential  ...  We use the following properties of matched and nested twins. Lemma 1.  ...

### GEM++: A Tool for Solving Substitution-Tolerant Subgraph Isomorphism [chapter]

Julien Lerouge, Pierre Le Bodic, Pierre Héroux, Sébastien Adam
2015 Lecture Notes in Computer Science
This paper describes an extended version of a Binary Linear Program (BLP) for solving this class of graph matching problem.  ...  The substitution-tolerant subgraph isomorphism is a particular error-tolerant subgraph matching that allows label substitutions for both vertices and edges.  ...  ), and the command-line utility GEM++sub, that solves our subgraph isomorphism problem.  ...

### What's Next? Future Directions in Parameterized Complexity [chapter]

Dániel Marx
2012 Lecture Notes in Computer Science
In particular, we set up a framework for understanding which problems can be solved by branching algorithms.  ...  Finally, we discuss that the domain of directed graph problems is a challenging area which can potentially see significant progress in the following years.  ...  A planar problem which is not fully understood yet is Subgraph Isomorphism: given graphs H and G, does G has a subgraph isomorphic to H?  ...

### Mine 'Em All: A Note on Mining All Graphs [chapter]

Ondřej Kuželka, Jan Ramon
2016 Lecture Notes in Computer Science
For instance, the problem can be solved with polynomial delay for databases of planar graphs when the enumerated graphs should be output from large to small but it cannot be solved even in incremental-polynomial  ...  We show that there are hereditary classes of graphs for which the complexity of this problem depends on the order in which the graphs should be enumerated (e.g. from frequent to infrequent or from small  ...  The problem ALL F →I cannot be solved in incremental polynomial time for the class G of planar graphs and subgraph isomorphism as pattern matching operator. Proof.  ...