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Martín D. Safe

Publications and source records attributed to Martín D. Safe.

At least 19 recordsLinked to original sources
arXiv2026

Characterization and linear-time recognition of balanced distance-hereditary graphs

A graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two $1$'s in each row and in each column. Although it is known that a graph is balanced if and only if it contains no induced extended odd sun, a characterization of balanced graphs by minimal forbidden induced subgraphs is still unknown. In this work, we prove that, within the class of distance-hereditary graphs, balanced graphs are exactly the hereditary clique-Helly graphs. Equivalently, they are characterized by a single forbidden induced subgraph, namely $\overline{3K_2}$. From this result, we derive an explicit linear-time algorithm that decides balancedness within the class of distance-hereditary graphs and returns an induced $\overline{3K_2}$ when the input graph is not balanced.

math.CO↗
arXiv2023

Nonexistence of uniformly most reliable graphs of least corank

If $G$ is a simple graph and $ρ\in[0,1]$, the reliability $R_G(ρ)$ is the probability of $G$ being connected after each of its edges is removed independently with probability $ρ$. A simple graph $G$ is a \emph{uniformly most reliable graph} (UMRG) if $R_G(ρ)\geq R_H(ρ)$ for every $ρ\in[0,1]$ and every simple graph $H$ on the same number of vertices and edges as $G$. Boesch [J.\ Graph Theory 10 (1986), 339--352] conjectured that, if $n$ and $m$ are such that there exists a connected simple graph on $n$ vertices and $m$ edges, then there also exists a UMRG on the same number of vertices and edges. Some counterexamples to Boesch's conjecture were given by Kelmans, Myrvold et al., and Brown and Cox. It is known that Boesch's conjecture holds whenever the corank, defined as $c=m-n+1$, is at most $4$ (and the corresponding UMRGs are fully characterized). Ath and Sobel conjectured that Boesch's conjecture holds whenever the corank $c$ is between $5$ and $8$, provided the number of vertices is at least $2c-2$. In this work, we give an infinite family of counterexamples to Boesch's conjecture of corank $5$. These are the first reported counterexamples that attain the minimum possible corank. As a byproduct, the conjecture by Ath and Sobel is disproved.

math.CO↗
arXiv2022

Lochs-type theorems beyond positive entropy

Lochs' theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. In its original version, Lochs' theorem related decimal expansions with continued fraction expansions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. This is the viewpoint we develop here. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature.

math.NT↗
arXiv2022

On the generalized Helly property of hypergraphs, cliques, and bicliques

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 $(p,q)$-the 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. We prove several characterizations of hereditary $(p,q)$-Helly hypergraphs, including one by minimal forbidden partial subhypergraphs. 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 but co-NP-complete if $p$ is part of the input. In addition, we generalize to $(p,q)$-clique-Helly graphs the characterization of $p$-clique-Helly graphs in terms of expansions 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 but NP-hard if $p$ or $q$ is part of the input. Finally, we provide different characterizations, give recognition algorithms, and prove hardness results for (hereditary) $(p,q)$-biclique-Helly graphs.

math.CO↗
arXiv2021

Essential obstacles to Helly circular-arc graphs

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.

math.CO↗
arXiv2021

2-nested matrices: towards understanding the structure of circle graphs

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 characterised in terms of forbidden submatrices by Tucker in 1972. Several graph classes were characterised by means of this property, including interval graphs and strongly chordal digraphs. In this work, we define and characterise 2-nested matrices, which are $(0,1)$-matrices with a variant of the C1P and for which there is also 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 characterise split graphs that are also circle by minimal forbidden induced subgraphs.

math.CO↗
arXiv2020

Forbidden induced subgraph characterization of circle graphs within split graphs

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.

cs.DM↗
arXiv2020

Fefferman-Stein inequalities for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree

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 the following Fefferman-Stein estimate \[ w\left(\left\{ x\in T\,:\,Mf(x)>λ\right\} \right)\leq c_{s}\frac{1}λ\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 \] is settled and moreover it {is shown it} is sharp, in the sense that it does not hold in general if $s=1$. Some examples of non trivial 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 on infinite trees is established.

math.CA↗
arXiv2020

Formulas in connection with parameters related to convexity of paths on three vertices: caterpillars and unit interval graphs

We present formulas to compute the P3-geodetic number, the P3-hull number and the percolation time for a caterpillar, in terms of certain sequences associated with it. In addition, we find a connection between the percolation time of a unit interval graph and a parameter involving the diameter of a unit interval graph related to it. Finally, we present a hereditary graph class, defined by forbidden induced subgraphs, such that its percolation time is equal to one.

math.CO↗
arXiv2019

On nested and 2-nested graphs: two subclasses of graphs between threshold and split graphs

A $(0,1)$-matrix has the Consecutive Ones Property (C1P) for the rows if there is a permutation of its columns such that the ones in each row appear consecutively. We say a $(0, 1)$-matrix is nested if it has the consecutive ones property for the rows (C1P) and every two rows are either disjoint or nested. We say a $(0, 1)$-matrix is 2-nested if it has the C1P and admits a partition of its rows into two sets such that the submatrix induced by each of these sets is nested. We say a split graph $G$ with split partition $(K, S)$ is nested (resp.\ 2-nested) if the matrix $A(S, K)$ which indicates the adjacency between vertices in $S$ and $K$ is nested (resp.\ 2-nested). In this work, we characterize nested and 2-nested matrices by minimal forbidden submatrices. This characterization leads to a minimal forbidden induced subgraph characterization for these classes of graphs, which are a superclass of threshold graphs and a subclass of split and circle graphs.

cs.DM↗
arXiv2019

Circularly compatible ones, $D$-circularity, and proper circular-arc bigraphs

In 1969, Alan 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 allowed him to devise the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as Tucker himself remarks, he 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 this 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, Das, Ghosh, and Sen [J. Graph Theory, 73(4):361--376, 2013]. Our findings generalize some known results about $D$-interval hypergraphs and proper interval bigraphs.

math.CO↗
arXiv2017

A $\{-1,0,1\}$- and sparsest basis for the null space of a forest in optimal time

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.

math.CO↗
arXiv2017

Partial characterization of graphs having a single large Laplacian eigenvalue

The parameter $σ(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $σ(G)=1$. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between $σ(G)$ and the number of anticomponents of $G$. As a by-product, we present some results which support the conjecture, by restricting our analysis to some classes of graphs.

math.CO↗
arXiv2016

Characterization and linear-time detection of minimal obstructions to concave-round graphs and the circular-ones property

A graph is concave-round if its vertices can be circularly enumerated so that the closed neighbourhood of each vertex is an interval in the enumeration. In this work, 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, 13:179--193, 2000]. 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.

math.CO↗
arXiv2016

Neighborhood covering and independence on two superclasses of cographs

Given a simple graph $G$, a set $C \subseteq V(G)$ is a neighborhood cover set if every edge and vertex of $G$ belongs to some $G[v]$ with $v \in C$, where $G[v]$ denotes the subgraph of $G$ induced by the closed neighborhood of the vertex $v$. Two elements of $E(G) \cup V(G)$ are neighborhood-independent if there is no vertex $v\in V(G)$ such that both elements are in $G[v]$. A set $S\subseteq V(G)\cup E(G)$ is neighborhood-independent if every pair of elements of $S$ is neighborhood-independent. Let $ρ_{\mathrm n}(G)$ be the size of a minimum neighborhood cover set and $α_{\mathrm n}(G)$ of a maximum neighborhood-independent set. Lehel and Tuza defined neighborhood-perfect graphs $G$ as those where the equality $ρ_{\mathrm n}(G') = α_{\mathrm n}(G')$ holds for every induced subgraph $G'$ of $G$. In this work we prove forbidden induced subgraph characterizations of the class of neighborhood-perfect graphs, restricted to two superclasses of cographs: $P_4$-tidy graphs and tree-cographs. We give as well linear-time algorithms for solving the recognition problem of neighborhood-perfect graphs and the problem of finding a minimum neighborhood cover set and a maximum neighborhood-independent set in these same classes.

cs.DM↗
arXiv2015

Graph classes with and without powers of bounded clique-width

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers $k$ and $\ell$ such that the $k$-th powers of the graphs are of clique-width at most $\ell$. 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.

math.CO↗
arXiv2014

Forbidden Induced Subgraphs of Normal Helly Circular-Arc Graphs: Characterization and Detection

A normal Helly circular-arc graph is the intersection graph of arcs on a circle of which no three or less arcs cover the whole circle. Lin, Soulignac, and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used it to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden induced 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 certificate.

cs.DM↗