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Interval graphs and proper interval graphs are well known graph classes, for which several generalizations have been proposed in the literature. In this work, we study the (proper) thinness, and several variations, for the classes of cographs, crowns graphs and grid graphs. We provide the exact values for several variants of thinness (proper, independent, complete, precedence, and combinations of them) for the crown graphs $CR_n$. For cographs, we prove that the precedence thinness can be determined in polynomial time. We also improve known bounds for the thinness of $n \times n$ grids $GR_n$ and $m \times n$ grids $GR_{m,n}$, proving that $\left \lceil \frac{n-1}{3} \right \rceil \leq \mbox{thin}(GR_n) \leq \left \lceil \frac{n+1}{2} \right \rceil$. Regarding the precedence thinness, we prove that $\mbox{prec-thin}(GR_{n,2}) = \left \lceil \frac{n+1}{2} \right \rceil$ and that $\left \lceil \frac{n-1}{3} \right \rceil \left \lceil\frac{n-1}{2} \right \rceil + 1 \leq \mbox{prec-thin}(GR_n) \leq \left \lceil\frac{n-1}{2} \right \rceil^2+1$. As applications, we show that the $k$-coloring problem is NP-complete for precedence $2$-thin graphs and for proper $2$-thin graphs, when $k$ is part of the input. On the positive side, it is polynomially solvable for precedence proper $2$-thin graphs, given the order and partition.

This paper presents an empirical performance analysis of three project scheduling algorithms dealing with maximizing projects' net present value with unrestricted resources. The selected algorithms, being the most recently cited in the literature, are: Recursive Search (RS), Steepest Ascent Approach (SAA) and Hybrid Search (HS). The main motivation for this research is the lack of knowledge about the computational complexities of the RS, SAA, and HS algorithms, since all studies to date show some gaps in the analysis. Furthermore, the empirical analysis performed to date does not consider the fact that one algorithm (HS) uses a dual search strategy, which markedly improved the algorithm's performance, while the others don't. In order to obtain a fair performance comparison, we implemented the dual search strategy into the other two algorithms (RS and SAA), and the new algorithms were called Recursive Search Forward-Backward (RSFB) and Steepest Ascent Approach Forward-Backward (SAAFB). The algorithms RSFB, SAAFB, and HS were submitted to a factorial experiment with three different project network sampling characteristics. The results were analyzed using the Generalized Linear Models (GLM) statistical modeling technique that showed: a) the general computational costs of RSFB, SAAFB, and HS; b) the costs of restarting the search in the spanning tree as part of the total cost of the algorithms; c) and statistically significant differences between the distributions of the algorithms' results.

A pendant vertex is one of degree one and an isolated vertex has degree zero. A neighborhood star-free (NSF for short) graph is one in which every vertex is contained in a triangle except pendant vertices and isolated vertices. This class has been considered before for several contexts. In the present paper, we study the complexity of the dominating induced matching (DIM) problem and the perfect edge domination (PED) problem for NSF graphs. We prove the corresponding decision problems are NP-Complete for several of its subclasses. As an added value of this study, we have shown three connected variants of planar positive 1in3SAT are also NP-Complete. Since these variants are more basic in complexity theory context than many graph problems, these results can be useful to prove that other problems are NP-Complete.

We introduce a new class of intersection graphs, the edge intersection graphs of paths on a triangular grid, called EPGt graphs. We show similarities and differences from this new class to the well-known class of EPG graphs. A turn of a path at a grid point is called a bend. An EPGt representation in which every path has at most $k$ bends is called a B$_k$-EPGt representation and the corresponding graphs are called B$_k$-EPGt graphs. We provide examples of B$_{2}$-EPG graphs that are B$_{1}$-EPGt. We characterize the representation of cliques with three vertices and chordless 4-cycles in B$_{1}$-EPGt representations. We also prove that B$_{1}$-EPGt graphs have Strong Helly number $3$. Furthermore, we prove that B$_{1}$-EPGt graphs are $7$-clique colorable., Comment: 19 pages, 12 figures

A matching $M$ is a $\mathscr{P}$-matching if the subgraph induced by the endpoints of the edges of $M$ satisfies property $\mathscr{P}$. As examples, for appropriate choices of $\mathscr{P}$, the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum $\mathscr{P}$-matching is a knowingly NP-Hard problem, with few exceptions, such as connected matchings, which has the same time complexity as usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent researches in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching $M$ such that the endpoint vertices of its edges induce a connected subgraph and the sum of the edge weights of $M$ is maximum. Unlike the unweighted Connected Matching problem, which is in P for general graphs, we show that Maximum Weight Connected Matching is NP-Hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite, and bounded degree planar graphs, while solvable in linear time for trees and subcubic graphs. When we restrict edge weights to be non negative only, we show that the problem turns to be polynomially solvable for chordal graphs, while it remains NP-Hard for most of the cases when weights can be negative. Our final contributions are on parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. In terms of kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover under standard complexity-theoretical hypotheses.

In 2005, Goddard, Hedetniemi, Hedetniemi and Laskar [Generalized subgraph-restricted matchings in graphs, Discrete Mathematics, 293 (2005) 129 - 138] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper we answer this question. In fact, we consider the generalized problem of finding $c$-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least $c$ connected components. We show that, for every fixed $c \geq 2$, this problem is NP-complete even if we restrict the input to bounded diameter bipartite graphs, while can be solved in polynomial time if $c = 1$. For the case when $c$ is part of the input, we show that the problem is NP-complete for chordal graphs, while being solvable in polynomial time for interval graphs. Finally, we explore the parameterized complexity of the problem. We present an FPT algorithm under the treewidth parameterization, and an XP algorithm for graphs with a polynomial number of minimal separators when parameterized by $c$. We complement these results by showing that, unless NP $\subseteq$ coNP/poly, the related Induced Matching problem does not admit a polynomial kernel when parameterized by vertex cover and size of the matching nor when parameterized by vertex deletion distance to clique and size of the matching. As for Connected Matching, we show how to obtain a maximum connected matching in linear time given an arbitrary maximum matching in the input.

The three-in-a-tree problem asks for an induced tree of the input graph containing three mandatory vertices. In 2006, Chudnovsky and Seymour [Combinatorica, 2010] presented the first polynomial time algorithm for this problem, which has become a critical subroutine in many algorithms for detecting induced subgraphs, such as beetles, pyramids, thetas, and even and odd-holes. In 2007, Derhy and Picouleau [Discrete Applied Mathematics, 2009] considered the natural generalization to $k$ mandatory vertices, proving that, when $k$ is part of the input, the problem is $\mathsf{NP}$-complete, and ask what is the complexity of four-in-a-tree. Motivated by this question and the relevance of the original problem, we study the parameterized complexity of $k$-in-a-tree. We begin by showing that the problem is $\mathsf{W[1]}$-hard when jointly parameterized by the size of the solution and minimum clique cover and, under the Exponential Time Hypothesis, does not admit an $n^{o(k)}$ time algorithm. Afterwards, we use Courcelle's Theorem to prove fixed-parameter tractability under cliquewidth, which prompts our investigation into which parameterizations admit single exponential algorithms; we show that such algorithms exist for the unrelated parameterizations treewidth, distance to cluster, and distance to co-cluster. In terms of kernelization, we present a linear kernel under feedback edge set, and show that no polynomial kernel exists under vertex cover nor distance to clique unless $\mathsf{NP} \subseteq \mathsf{coNP}/\mathsf{poly}$. Along with other remarks and previous work, our tractability and kernelization results cover many of the most commonly employed parameters in the graph parameter hierarchy., Comment: 25 pages, 8 figures

Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of $k$ interval graphs, by splitting the graph in some special way. As a recent example of such an approach, the classes of $k$-thin and proper $k$-thin graphs have been introduced generalizing interval and proper interval graphs, respectively. The complexity of the recognition of each of these classes is still open, even for fixed $k \geq 2$. In this work, we introduce a subclass of $k$-thin graphs (resp. proper $k$-thin graphs), called precedence $k$-thin graphs (resp. precedence proper $k$-thin graphs). Concerning partitioned precedence $k$-thin graphs, we present a polynomial time recognition algorithm based on $PQ$-trees. With respect to partitioned precedence proper $k$-thin graphs, we prove that the related recognition problem is \NP-complete for an arbitrary $k$ and polynomial-time solvable when $k$ is fixed. Moreover, we present a characterization for these classes based on threshold graphs., Comment: 33 pages

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function., Comment: 45 pages. arXiv admin note: text overlap with arXiv:1704.00379

Since many NP-complete graph problems have been shown polynomial-time solvable when restricted to claw-free graphs, we study the problem of determining the distance of a given graph to a claw-free graph, considering vertex elimination as measure. CLAW-FREE VERTEX DELETION (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is claw-free. Although CFVD is NP-complete in general and recognizing claw-free graphs is still a challenge, where the current best algorithm for a graph $G$ has the same running time of the best algorithm for matrix multiplication, we present linear-time algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem can be solved in linear time by a simpler algorithm on forests, and we determine the exact values for full $k$-ary trees. On the other hand, we show that CLAW-FREE VERTEX DELETION is NP-complete even when the input graph is a split graph. We also show that the problem is hard to approximate within any constant factor better than $2$, assuming the Unique Games Conjecture., Comment: 20 pages

Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. In this paper, we show that given a graph $G$ and an integer $k$, the problem of determining whether $G$ admits a $B_k$-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. Moreover, we show that the problem of recognizing Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.

Let $G$ be a finite, simple, and undirected graph and let $S$ be a set of vertices of $G$. In the geodetic convexity, a set of vertices $S$ of a graph $G$ is convex if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The convex hull $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a hull set. The cardinality $h(G)$ of a minimum hull set of $G$ is the hull number of $G$. The complementary prism $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. Motivated by previous work, we determine and present lower and upper bounds on the hull number of complementary prisms of trees, disconnected graphs and cographs. We also show that the hull number on complementary prisms cannot be limited in the geodetic convexity, unlike the $P_3$-convexity., Comment: 12 pages, 5 figures

We study the problem of determining whether a given graph~$G=(V,E)$ admits a matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently whether~$G-M$ is bipartite). This problem is equivalent to determine whether~$G$ admits a~$(2,1)$-coloring, which is a~$2$-coloring of~$V(G)$ such that each color class induces a graph of maximum degree at most~$1$. We determine a dichotomy related to the~{\sf NP}-completeness of this problem, where we show that it is~{\sf NP}-complete even for $3$-colorable planar graphs of maximum degree~$4$, while it is known that the problem can be solved in polynomial time for graphs of maximum degree at most~$3$. In addition we present polynomial-time algorithms for some graph classes, including graphs in which every odd cycle is a triangle, graphs of small dominating sets, and~$P_5$-free graphs. Additionally, we show that the problem is fixed parameter tractable when parameterized by the clique-width, which implies polynomial-time solution for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs. Finally, an~$O\left(2^{O\left(vc(G)\right)} \cdot n\right)$-time algorithm and a kernel of at most~$2\cdot nd(G)$ vertices are presented, where~$vc(G)$ and~$nd(G)$ are the vertex cover number and the neighborhood diversity of~$G$, respectively., Comment: A conference version appeared in the Proc. of the 12th Annual International Conference on Combinatorial Optimization and Applications (COCOA), Volume 11346, pages 198--213, Atlanta, USA, December 2018

Let $G$ be an undirected graph. An edge of $G$ dominates itself and all edges adjacent to it. A subset $E'$ of edges of $G$ is an edge dominating set of $G$, if every edge of the graph is dominated by some edge of $E'$. We say that $E'$ is a perfect edge dominating set of $G$, if every edge not in $E'$ is dominated by exactly one edge of $E'$. The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of $G$. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most $d \geq 3$ and large girth. In contrast, we prove that the problem admits an $O(n)$ time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a $P_5$-free graph. The algorithm is robust, in the sense that, given an arbitrary graph $G$, either it computes a minimum weight perfect edge dominating set of $G$, or it exhibits an induced subgraph of $G$, isomorphic to a $P_5$.

We revisit a concept that has been central in some early stages of computer science, that of structured programming: a set of rules that an algorithm must follow in order to acquire a structure that is desirable in many aspects. While much has been written about structured programming, an important issue has been left unanswered: given an arbitrary, compiled program, describe an algorithm to decide whether or not it is structured, that is, whether it conforms to the stated principles of structured programming. We refer to the classical concept of structured programming, as described by Dijkstra. By employing a graph model and graph-theoretic techniques, we formulate an efficient algorithm for answering this question. To do so, we first introduce the class of graphs which correspond to structured programs, which we call Dijkstra Graphs. Our problem then becomes the recognition of such graphs, for which we present a greedy $O(n)$-time algorithm. Furthermore, we describe an isomorphism algorithm for Dijkstra graphs, whose complexity is also linear in the number of vertices of the graph. Both the recognition and isomorphism algorithms have potential important applications, such as in code similarity analysis., Comment: 20 pages, 9 figures

A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if and only if $|S(u)\cap S(v)|\geq p$. We show that every minimal forbidden induced subgraph for the hereditary class ${\cal G}(d,p)$ of graphs whose $p$-intersection number is at most $d$, has order at most $3\cdot 2^{d+1}+1$, and that the exponential dependence on $d$ in this upper bound is necessary. For $p\in \{ d-1,d-2\}$, we provide more explicit results characterizing the graphs in ${\cal G}(d,p)$ without isolated/universal vertices using forbidden induced subgraphs.

Given a graph $G = (V,E)$, a \emph{perfect dominating set} is a subset of vertices $V' \subseteq V(G)$ such that each vertex $v \in V(G)\setminus V'$ is dominated by exactly one vertex $v' \in V'$. An \emph{efficient dominating set} is a perfect dominating set $V'$ where $V'$ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remains NP-Hard, even when restricted to certain graphs families. We study both variants of the problems for the circular-arc graphs, and show efficient algorithms for all of them.

Say that an edge of a graph $G$ dominates itself and every other edge adjacent to it. An edge dominating set of a graph $G=(V,E)$ is a subset of edges $E' \subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is dominated by exactly one edge of $E'$ then $E'$ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of finding a minimum weighted dominating induced matching, if any, and counting the number of dominating induced matchings of a graph with weighted edges. We describe an exact algorithm for general graphs that runs in $O^*(1.1939^n)$ time and polynomial (linear) space. This improves over any existing exact algorithm for the problems in consideration., Comment: 17 pages

Digital watermarks have been considered a promising way to fight software piracy. Graph-based watermarking schemes encode authorship/ownership data as control-flow graph of dummy code. In 2012, Chroni and Nikolopoulos developed an ingenious such scheme which was claimed to withstand attacks in the form of a single edge removal. We extend the work of those authors in various aspects. First, we give a formal characterization of the class of graphs generated by their encoding function. Then, we formulate a linear-time algorithm which recovers from ill-intentioned removals of $k \leq 2$ edges, therefore proving their claim. Furthermore, we provide a simpler decoding function and an algorithm to restore watermarks with an arbitrary number of missing edges whenever at all possible. By disclosing and improving upon the resilience of Chroni and Nikolopoulos's watermark, our results reinforce the interest in regarding it as a possible solution to numerous applications., Comment: 44 pages, 6 figures. An extended abstract of this paper was published in Proceedings of the 39th International Workshop on Graph Theoretic Concepts in Computer Science (WG 2013), Lecture Notes in Computer Science 8165 (2013), 50-63

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of E' then E' is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit is NP-complete. In this paper we consider the problem of finding a minimum weighted dominating induced matching, if any, of a graph with weighted edges. We describe two exact algorithms for general graphs. The algorithms are efficient in the cases where G admits a known vertex dominating set of small size, or when G contains a polynomial number of maximal independent sets., Comment: 9 pages

We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs., Comment: 25 pages, 9 figures

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs., Comment: 39 pages, 13 figures. A previous version of the paper (entitled Proper Helly Circular-Arc Graphs) appeared at WG'07

In this paper we present a modification of a technique by Chiba and Nishizeki [Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J. Comput. 14(1), pp. 210--223 (1985)]. Based on it, we design a data structure suitable for dynamic graph algorithms. We employ the data structure to formulate new algorithms for several problems, including counting subgraphs of four vertices, recognition of diamond-free graphs, cop-win graphs and strongly chordal graphs, among others. We improve the time complexity for graphs with low arboricity or h-index., Comment: 19 pages, no figures