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A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are precisely the minimal forbidden induced circular-arc subgraphs for the class of Helly circular-arc graphs. We show that it is possible to find in linear time, in any given obstacle, some minimal forbidden induced subgraph for the class of Helly circular-arc graphs contained as an induced subgraph. Moreover, relying on an existing linear-time algorithm for finding induced obstacles in circular-arc graphs, we conclude that it is possible to find in linear time an induced essential obstacle in any circular-arc graph that is not a Helly circular-arc graph. The problem of finding a forbidden induced subgraph characterization, not restricted only to circular-arc graphs, for the class of Helly circular-arc graphs remains unresolved. As a partial answer to this problem, we find the minimal forbidden induced subgraph characterization for the class of Helly circular-arc graphs restricted to graphs containing no induced claw and no induced 5-wheel. Furthermore, we show that there is a linear-time algorithm for finding, in any given graph that is not a Helly circular-arc graph, an induced subgraph isomorphic to claw, 5-wheel, or some minimal forbidden induced subgraph for the class of Helly circular-arc graphs. [ABSTRACT FROM AUTHOR]

In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results led to the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as remarked there, this work did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). In the present work, we solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related D-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu et al. [J. Graph Theory, 73 (2013), pp. 361--376]. Our findings generalize some known results about D-interval hypergraphs and proper interval bigraphs. [ABSTRACT FROM AUTHOR]

A graph is concave‐round if its vertices can be circularly enumerated so that the closed neighborhood of each vertex is an interval in the enumeration. In this study, we give a minimal forbidden induced subgraph characterization for the class of concave‐round graphs, solving a problem posed by Bang‐Jensen, Huang, and Yeo [SIAM J. Discrete Math., 13 (2000), pp. 179–193]. In addition, we show that it is possible to find one such forbidden induced subgraph in linear time in any given graph that is not concave‐round. As part of the analysis, we obtain characterizations by minimal forbidden submatrices for the circular‐ones property for rows and for the circular‐ones property for rows and columns and show that, also for both variants of the property, one of the corresponding forbidden submatrices can be found (if present) in any given matrix in linear time. We make some final remarks regarding connections to some classes of circular‐arc graphs. [ABSTRACT FROM AUTHOR]

A family of sets is (p , q) - intersecting if every nonempty subfamily of p or fewer sets has at least q elements in its total intersection. A family of sets has the (p , q) - Helly property if every nonempty (p , q) -intersecting subfamily has total intersection of cardinality at least q. The (2 , 1) -Helly property is the usual Helly property. A hypergraph is (p , q) - Helly if its edge family has the (p , q) -Helly property and hereditary (p , q) -Helly if each of its subhypergraphs has the (p , q) -Helly property. A graph is (p , q) - clique-Helly if the family of its maximal cliques has the (p , q) -Helly property and hereditary (p , q) -clique-Helly if each of its induced subgraphs is (p , q) -clique-Helly. The classes of (p , q) - biclique-Helly and hereditary (p , q) -biclique-Helly graphs are defined analogously. In this work, we prove several characterizations of hereditary (p , q) -Helly hypergraphs, including one by minimal forbidden partial subhypergraphs. On the algorithmic side, we give an improved time bound for the recognition of (p , q) -Helly hypergraphs for each fixed q and show that the recognition of hereditary (p , q) -Helly hypergraphs can be solved in polynomial time if p and q are fixed and co-NP-complete if p is part of the input. In addition, we generalize the characterization of p -clique-Helly graphs in terms of expansions to (p , q) -clique-Helly graphs and give different characterizations of hereditary (p , q) -clique-Helly graphs, including one by forbidden induced subgraphs. We give an improvement on the time bound for the recognition of (p , q) -clique-Helly graphs and prove that the recognition problem of hereditary (p , q) -clique-Helly graphs is polynomial-time solvable for p and q fixed and NP-hard if p or q is part of the input. Finally, we provide different characterizations, give recognition algorithms, and prove hardness results for (p , q) -biclique-Helly graphs and hereditary (p , q) -biclique-Helly graphs which are analogous to those for (p , q) -clique-Helly and hereditary (p , q) -clique-Helly graphs. [ABSTRACT FROM AUTHOR]

A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs. [ABSTRACT FROM AUTHOR]

A (0, 1)-matrix has the consecutive-ones property (C1P) if its columns can be permuted to make the 1’s in each row appear consecutively. This property was characterized in terms of forbidden submatrices by Tucker in 1972. Several graph classes were characterized by means of this property, including interval graphs and strongly chordal digraphs. In this work, we define and characterize 2-nested matrices, which are (0, 1)-matrices with a variant of the C1P and for which there is also a certain assignment of one of two colors to each block of consecutive 1’s in each row. The characterization of 2-nested matrices in the present work is of key importance to characterize split graphs that are also circle by minimal forbidden induced subgraphs. [ABSTRACT FROM AUTHOR]

Let G be a connected graph on n vertices and D (G) be its distance matrix. The formula for computing the determinant of this matrix in terms of the number of vertices is known when the graph is either a tree or a unicyclic graph. In this work we generalize these results, obtaining the determinant of the distance matrix of any graph whose block decomposition consists of edges, unicyclic and bicyclic graphs. [ABSTRACT FROM AUTHOR]

In this paper, weighted endpoint estimates for the Hardy–Littlewood maximal function on the infinite rooted |$k$| -ary tree are provided. Motivated by Naor and Tao [ 23 ], the following Fefferman–Stein estimate $$\begin{align*}& w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}\: \text{d}x\qquad s>1\end{align*}$$ is settled, and moreover, it is shown that it is sharp, in the sense that it does not hold in general if |$s=1$|⁠. Some examples of nontrivial weights such that the weighted weak type |$(1,1)$| estimate holds are provided. A strong Fefferman–Stein-type estimate and as a consequence some vector-valued extensions are obtained. In the appendix, a weighted counterpart of the abstract theorem of Soria and Tradacete [ 38 ] on infinite trees is established. [ABSTRACT FROM AUTHOR]

Perfect graphs form a well-known class of graphs introduced by Berge in the 1960s in terms of a min–max type equality involving two famous graph parameters. In this work, we survey certain variants and subclasses of perfect graphs defined by means of min–max relations of other graph parameters; namely: clique-perfect, coordinated, and neighborhood-perfect graphs. We show the connection between graph classes and both hypergraph theory, the clique graph operator, and some other graph classes. We review different partial characterizations of them by forbidden induced subgraphs, present the previous results, and the main open problems. Computational complexity problems are also discussed. [ABSTRACT FROM AUTHOR]

A normal Helly circular-arc graph is the intersection graph of a set of arcs on a circle of which no three or less arcs cover the whole circle. Lin et al. (2013) characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used them to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden subgraph characterization and a direct recognition algorithm for normal Helly circular-arc graphs, both of which are resolved by the current paper. Moreover, when the input is not a normal Helly circular-arc graph, our recognition algorithm finds in linear time a minimal forbidden induced subgraph as a certificate. Our approach yields also a considerably simpler algorithm for the certifying recognition of proper Helly circular-arc graphs, a subclass of normal Helly circular-arc graphs. [ABSTRACT FROM AUTHOR]

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there are diverse characterizations of circle graphs, a structural characterization by minimal forbidden induced subgraphs for the entire class of circle graphs is not known, not even restricted to split graphs (which are the graphs whose vertex set can be partitioned into a clique and a stable set). In this work, we give a characterization by minimal forbidden induced subgraphs of circle graphs, restricted to split graphs. [ABSTRACT FROM AUTHOR]

A set of vertices X of a graph G is convex if no shortest path between two vertices in X contains a vertex outside X . We prove that for fixed p ≥ 1 , all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time. [ABSTRACT FROM AUTHOR]

A graph is clique-perfect if the maximum number of pairwise disjoint maximal cliques equals the minimum number of vertices intersecting all maximal cliques for each induced subgraph. In this work, we give necessary and sufficient conditions for the complement of a line graph to be clique-perfect and an O ( n 2 ) -time algorithm to recognize such graphs. These results follow from a characterization and a linear-time recognition algorithm for matching-perfect graphs, which we introduce as graphs where the maximum number of pairwise edge-disjoint maximal matchings equals the minimum number of edges intersecting all maximal matchings for each subgraph. Thereby, we completely describe the linear and circular structure of the graphs containing no bipartite claw, from which we derive a structure theorem for all those graphs containing no bipartite claw that are Class 2 with respect to edge-coloring. [ABSTRACT FROM AUTHOR]

A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness ofP4-tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for anyP4-tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced. [ABSTRACT FROM PUBLISHER]

Abstract: Circular-arc graphs are the intersection graphs of open arcs on a circle. Circle graphs are the intersection graphs of chords on a circle. These graph classes have been the subject of much study for many years and numerous interesting results have been reported. Many subclasses of both circular-arc graphs and circle graphs have been defined and different characterizations formulated. In this survey, we summarize the most important structural results related to circular-arc graphs and circle graphs and present the main open problems. [Copyright &y& Elsevier]

Abstract: A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes. [Copyright &y& Elsevier]

Given a matrix, the Null Space Problem asks for a basis of its null space having the fewest nonzeros. This problem is known to be NP-complete and even hard to approximate. The null space of a forest is the null space of its adjacency matrix. Sander and Sander (2005) and Akbari et al. (2006), independently, proved that the null space of each forest admits a { − 1 , 0 , 1 } -basis. We devise an algorithm for determining a sparsest basis of the null space of any given forest which, in addition, is a { − 1 , 0 , 1 } -basis. Our algorithm is time-optimal in the sense that it takes time at most proportional to the number of nonzeros in any sparsest basis of the null space of the input forest. Moreover, we show that, given a forest F on n vertices, the set of those vertices x for which there is a vector in the null space of F that is nonzero at x and the number of nonzeros in any sparsest basis of the null space of F can be found in O ( n ) time. [ABSTRACT FROM AUTHOR]

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2 -domination number, γ w 2 ( G ) , the 2 -domination number, γ 2 ( G ) , the { 2 } -domination number, γ { 2 } ( G ) , the double domination number, γ × 2 ( G ) , the total { 2 } -domination number, γ t { 2 } ( G ) , and the total double domination number, γ t × 2 ( G ) , where G is a graph in which the corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γ R ( G ) , and two classical parameters, the domination number, γ ( G ) , and the total domination number, γ t ( G ) , we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs. [ABSTRACT FROM AUTHOR]

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers k and ℓ such that the k th powers of the graphs are of clique-width at most ℓ . We give sufficient and necessary conditions for this property. As our main results, we characterize graph classes of power-bounded clique-width within classes defined by either one forbidden induced subgraph, or by two connected forbidden induced subgraphs. We also show that for every positive integer k , there exists a graph class such that the k th powers of graphs in the class form a class of bounded clique-width, while this is not the case for any smaller power. [ABSTRACT FROM AUTHOR]

The matching number of a graph is the maximum size of a set of vertex-disjoint edges. The transversal number is the minimum number of vertices needed to meet every edge. A graph has the König–Egerváry property if its matching number equals its transversal number. Lovász proved a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs within graphs with a perfect matching. Korach, Nguyen, and Peis proposed an extension of Lovász’s result to a characterization of all graphs having the König–Egerváry property in terms of forbidden configurations (which are certain arrangements of a subgraph and a maximum matching). In this work, we prove a characterization of graphs having the König–Egerváry property by means of forbidden subgraphs which is a strengthened version of the characterization by Korach et al. Using our characterization of graphs with the König–Egerváry property, we also prove a forbidden subgraph characterization for the class of edge-perfect graphs. [ABSTRACT FROM AUTHOR]

A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe 풢) graphs with disconnected complement for every graph class 풢 with a companion. [ABSTRACT FROM AUTHOR]

Abstract: A circle graph is the intersection graph of a family of chords on a circle. There is no known characterization of circle graphs by forbidden induced subgraphs that do not involve the notions of local equivalence or pivoting operations. We characterize circle graphs by a list of minimal forbidden induced subgraphs when the graph belongs to one of the following classes: linear domino graphs, -tidy graphs, and tree-cographs. We also completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property. [Copyright &y& Elsevier]