A Polynomial Kernel for Line Graph Deletion

The line graph of a graph $G$ is the graph $L(G)$ whose vertex set is the edge set of $G$ and there is an edge between $e,f\in E(G)$ if $e$ and $f$ share an endpoint in $G$. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most $k$ edges from the input graph $G$ such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with $\mathcal{O}(k^{5})$ vertices. This answers an open question posed by Falk Huffner at Workshop on Kernels (WorKer) in 2013.

[1]  Paul D. Seymour,et al.  Claw-free graphs. IV. Decomposition theorem , 2008, J. Comb. Theory, Ser. B.

[2]  Reinhard Diestel,et al.  Graph Theory, 4th Edition , 2012, Graduate texts in mathematics.

[3]  Leizhen Cai,et al.  Incompressibility of $$H$$H-Free Edge Modification Problems , 2014, Algorithmica.

[4]  John M. Lewis,et al.  The Node-Deletion Problem for Hereditary Properties is NP-Complete , 1980, J. Comput. Syst. Sci..

[5]  L. Beineke.  Characterizations of derived graphs , 1970 .

[6]  R. B. Sandeep,et al.  Incompressibility of H-free edge modification problems: Towards a dichotomy , 2020, ESA.

[7]  H. L. Abbott,et al.  Intersection Theorems for Systems of Sets , 1972, J. Comb. Theory, Ser. A.

[8]  H. Whitney.  Congruent Graphs and the Connectivity of Graphs , 1932 .

[9]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[10]  Erik Jan van Leeuwen,et al.  Polynomial Kernelization for Removing Induced Claws and Diamonds , 2015, Theory of Computing Systems.

[11]  Naveen Sivadasan,et al.  Dichotomy Results on the Hardness of H-free Edge Modification Problems , 2017, SIAM J. Discret. Math..

[12]  Christophe Paul,et al.  On the (Non-)Existence of Polynomial Kernels for Pl-Free Edge Modification Problems , 2010, Algorithmica.

[13]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[14]  Leizhen Cai,et al.  Fixed-Parameter Tractability of Graph Modification Problems for Hereditary Properties , 1996, Inf. Process. Lett..

[15]  Eduard Eiben,et al.  A Polynomial Kernel for Paw-Free Editing , 2019, IPEC.

[16]  Charles J. Colbourn,et al.  The complexity of some edge deletion problems , 1988 .

[17]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[18]  Najiba Sbihi,et al.  Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile , 1980, Discret. Math..

[19]  Stefan Kratsch,et al.  Two edge modification problems without polynomial kernels , 2009, Discret. Optim..

[20]  Michael R. Fellows,et al.  Open Problems in Parameterized and Exact Computation - IWPEC 2006 , 2006 .

[21]  Klaus Simon,et al.  A Dynamic Algorithm for Line Graph Recognition , 1995, WG.

[22]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[23]  Leizhen Cai,et al.  Incompressibility of H-Free Edge Modification , 2013, IPEC.

[24]  Mihalis Yannakakis,et al.  Edge-Deletion Problems , 1981, SIAM J. Comput..