On recognizing words that are squares for the shuffle product

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On the complexity of Dominating Set for graphs with fixed diameter

A set $S\subseteq V$ of a graph $G=(V,E)$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Dominating Set is the problem of deciding, given a graph $G$ and an integer $k\geq 1$, if $G$ has a dominating set of size at most $k$. It is well known that this problem is $\mathsf{NP}$-complete even for claw-free graphs. We give a complexity dichotomy for Dominating Set for the class of claw-free graphs with diameter $d$. We show that the problem is $\mathsf{NP}$-complete for every fixed $d\ge 3$ and polynomial time solvable for $d\le 2$. To prove the case $d=2$, we show that Minimum Maximal Matching can be solved in polynomial time for $2K_2$-free graphs.Comment: 15 pages, 5 figure

Efficient algorithms for tuple domination on co-biconvex graphs and web graphs

A vertex in a graph dominates itself and each of its adjacent vertices. The $k$-tuple domination problem, for a fixed positive integer $k$, is to find a minimum sized vertex subset in a given graph such that every vertex is dominated by at least k vertices of this set. From the computational point of view, this problem is NP-hard. For a general circular-arc graph and $k=1$, efficient algorithms are known to solve it (Hsu et al., 1991 & Chang, 1998) but its complexity remains open for $k\geq 2$. A $0,1$-matrix has the consecutive 0's (circular 1's) property for columns if there is a permutation of its rows that places the 0's (1's) consecutively (circularly) in every column. Co-biconvex (concave-round) graphs are exactly those graphs whose augmented adjacency matrix has the consecutive 0's (circular 1's) property for columns. Due to A. Tucker (1971), concave-round graphs are circular-arc. In this work, we develop a study of the $k$-tuple domination problem on co-biconvex graphs and on web graphs which are not comparable and, in particular, all of them concave-round graphs. On the one side, we present an $O(n^2)$-time algorithm for solving it for each $2\leq k\leq |U|+3$, where $U$ is the set of universal vertices and $n$ the total number of vertices of the input co-biconvex graph. On the other side, the study of this problem on web graphs was already started by Argiroffo et al. (2010) and solved from a polyhedral point of view only for the cases $k=2$ and $k=d(G)$, where $d(G)$ equals the degree of each vertex of the input web graph $G$. We complete this study for web graphs from an algorithmic point of view, by designing a linear time algorithm based on the modular arithmetic for integer numbers. The algorithms presented in this work are independent but both exploit the circular properties of the augmented adjacency matrices of each studied graph class.Comment: 21 pages, 7 figures. Keywords: $k$-tuple dominating sets, augmented adjacency matrices, stable sets, modular arithmeti

Large-scale clique cover of real-world networks

The edge clique cover (ECC ) problem deals with discovering a set of (possibly overlapping) cliques in a given graph that covers each of the graph's edges. This problem finds applications ranging from social networks to compiler optimization and stringology. We consider several variants of the ECC problem, using classical quality measures (like the number of cliques) and new ones. We describe efficient heuristic algorithms, the fastest one taking O(mdG) time for a graph with m edges, degeneracy dG (also known as k-core number). For large real-world networks with millions of nodes, like social networks, an algorithm should have (almost) linear running time to be practical: Our algorithm for finding ECCs of large networks has linear-time performance in practice because dG is small, as our experiments show, on real-world networks with thousands to several million nodes

Maximum Independent Set And Maximum Induced Matching Problems For Competitive Programming

Competitive programming is a growing interest among students, with some students training for years to be competitive in national and international competitions. Competitive programming problems continue to become more complex; yet they are always solvable with skills learned in an undergraduate algorithms class. This makes competitive programming a great way for undergraduates to develop their coding skills and learn complex algorithms. However, there are very few competitive programming problems on particular graph classes, despite the fact that the field of graph theory is rich with complexity and algorithms results for over eighty years. This may be because of the overwhelming amount of graph classes and terminology that students need to be familiar with to understand even the simplest results in graph theory, sometimes overlooking the connection between graph theory and the study of algorithms. Some of these algorithms, like that of computing a maximum independent set (MIS) or a maximum induced matching (MIM) on special graph classes, only require techniques learned in an undergraduate algorithms course. However, in the literature, they are hidden behind results for generalized classes, often using terminology and notation far beyond what undergraduate students are exposed to. Some of these graph theoretic results are either so old that the original papers are hard to find or they are held behind payment gateways from publishers. Therefore, there needs to be substantive work done to improve the expositions of old (and some new) graph theory algorithms that can be solved using topics learned in an undergraduate course. This will allow students in algorithm classes to be exposed to topics in graph theory, while fundamental problems on graphs can be used as excellent motivating examples for topics in algorithms