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This paper deals with optimization problems on rotagraphs and fasciagraphs. These graphs are repetitive structures that generalize grids and toroidal grids, respectively. We develop a theoretical framework and get linear-time and constant-time generic algorithms for wide classes of combinatorial problems which are defined in a purely combinatorial way. These classes of problems include in particular many classical optimization problems that are NP-complete in general. Our results unify in a single framework many results of the literature, in particular on the topic of domination–and its variants–in grids. [ABSTRACT FROM AUTHOR]

A many-to-many k -disjoint path cover ( k -DPC for short) of a graph G joining the pairwise disjoint vertex sets S and T , each of size k , is a collection of k vertex-disjoint paths between S and T , which altogether cover every vertex of G . This is classified as paired , if each vertex of S must be joined to a specific vertex of T , or unpaired , if there is no such constraint. In this paper, we develop a linear-time algorithm for the Unpaired DPC problem of finding an unpaired DPC joining S and T given in a unit interval graph, to which the One-to-One DPC and the One-to-Many DPC problems are reduced in linear time. Additionally, we show that the Paired k - DPC problem on a unit interval graph is polynomially solvable for any fixed k . [ABSTRACT FROM AUTHOR]

The average eccentricity e c c ( G ) of a graph G is the mean value of eccentricities of all vertices of G . In this paper, we continue the work of Ilić (2012) and resolve a conjecture, obtained by the system AutoGraphiX, about the upper bound on the sum of the average eccentricity and the domination number among connected graphs on n vertices. [ABSTRACT FROM AUTHOR]

A subset F of vertices of a graph G is called a vertex cover P t ( V C P t ) set if every path of order t in G contains at least one vertex from F . The vertex cover P t ( V C P t ) problem is to find a minimum V C P t set in a graph. The V C P t problem is NP-complete for any integer t ≥ 2 . In this paper, we restrict our attention to the V C P 4 problem and present an FPT algorithm with runtime O ∗ ( 3 k ) for the V C P 4 problem. The algorithm constructs a V C P 4 set of size at most k in a given graph G , or reports that no such V C P 4 set exists in G . [ABSTRACT FROM AUTHOR]

A dominating induced matching , also called an efficient edge domination , of a graph G = ( V , E ) with n = | V | vertices and m = | E | edges is a subset F ⊆ E of edges in the graph such that no two edges in F share a common endpoint and each edge in E ∖ F is incident with exactly one edge in F . It is NP-hard to decide whether a graph admits a dominating induced matching or not. In this paper, we design a 1.146 7 n n O ( 1 ) -time exact algorithm for this problem, improving all previous results. This problem can be redefined as a partition problem that is to partition the vertex set of a graph into two parts I and F , where I induces an independent set (a 0-regular graph) and F induces a perfect matching (a 1-regular graph). After giving several structural properties of the problem, we show that the problem always contains some “good vertices,” branching on which by including them to either I or F we can effectively reduce the graph. This leads to a fast exact algorithm to this problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R 2 . First, we propose an O ( n 2 log n ) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangles, the time complexity can be slightly improved to O ( n log n + n K ) , where K is the size of the maximum clique. For disk graphs, we consider two variations of the maximum clique problem, namely geometric clique and graphical clique. The time complexity of our algorithm for finding the largest geometric clique is O ( m log n + n 2 ) where m is the number of edges in the disk graph, and it works for disks of arbitrary radii. For graphical clique, our proposed algorithm works for unit disks (i.e., of same radii) and the worst case time complexity is O ( n 2 + m ( n + K 3 ) ) ; m is the number of edges in the unit disk intersection graph and K is the size of the largest clique in that graph. It uses O ( n 3 ) time in-place computation of maximum matching in a bipartite graph, where the vertices are given in an array, and the existence of an edge between a pair of vertices can be checked by an oracle on demand (from problem specification) in O ( 1 ) time. This problem is of independent interest. All these algorithms need O ( 1 ) work space in addition to the input array. [ABSTRACT FROM AUTHOR]