arXiv subjects

Tiziana Calamoneri

Publications and source records attributed to Tiziana Calamoneri.

16 recordsLinked to original sources

arXiv2025

m-Eternal Domination and Variants on Some Classes of Finite and Infinite Graphs

We study the m-Eternal Domination problem, which is the following two-player game between a defender and an attacker on a graph: initially, the defender positions k guards on vertices of the graph; the game then proceeds in turns between the defender and the attacker, with the attacker selecting a vertex and the defender responding to the attack by moving a guard to the attacked vertex. The defender may move more than one guard on their turn, but guards can only move to neighboring vertices. The defender wins a game on a graph G with k guards if the defender has a strategy such that at every point of the game the vertices occupied by guards form a dominating set of G and the attacker wins otherwise. The m-eternal domination number of a graph G is the smallest value of k for which (G,k) is a defender win. We show that m-Eternal Domination is NP-hard, as well as some of its variants, even on special classes of graphs. We also show structural results for the Domination and m-Eternal Domination problems in the context of four types of infinite regular grids: square, octagonal, hexagonal, and triangular, establishing tight bounds.

cs.DM↗

arXiv2025

The Minimum Eternal Vertex Cover Problem on a Subclass of Series-Parallel Graphs

Eternal vertex cover is the following two-player game between a defender and an attacker on a graph. Initially, the defender positions k guards on k vertices of the graph; the game then proceeds in turns between the defender and the attacker, with the attacker selecting an edge and the defender responding to the attack by moving some of the guards along the edges, including the attacked one. The defender wins a game on a graph G with k guards if they have a strategy such that, in every round of the game, the vertices occupied by the guards form a vertex cover of G, and the attacker wins otherwise. The eternal vertex cover number of a graph G is the smallest number k of guards allowing the defender to win and Eternal Vertex Cover is the problem of computing the eternal vertex cover number of the given graph. We study this problem when restricted to the well-known class of series-parallel graphs. In particular, we prove that Eternal Vertex Cover can be solved in linear time when restricted to melon graphs, a proper subclass of series-parallel graphs. Moreover, we also conjecture that this problem is NP-hard on series-parallel graphs.

math.CO↗

arXiv2024

All Graphs with at most 8 nodes are 2-interval-PCGs

A graph G is a multi-interval PCG if there exist an edge weighted tree T with non-negative real values and disjoint intervals of the non-negative real half-line such that each node of G is uniquely associated to a leaf of T and there is an edge between two nodes in G if and only if the weighted distance between their corresponding leaves in T lies within any such intervals. If the number of intervals is k, then we call the graph a k-interval-PCG; in symbols, G = k-interval-PCG (T, I1, . . . , Ik). It is known that 2-interval-PCGs do not contain all graphs and the smallest known graph outside this class has 135 nodes. Here we prove that all graphs with at most 8 nodes are 2-interval-PCGs, so doing one step towards the determination of the smallest value of n such that there exists an n node graph that is not a 2-interval-PCG.

cs.DM↗

arXiv2024

Toward a Unified Graph-Based Representation of Medical Data for Precision Oncology Medicine

We present a new unified graph-based representation of medical data, combining genetic information and medical records of patients with medical knowledge via a unique knowledge graph. This approach allows us to infer meaningful information and explanations that would be unavailable by looking at each data set separately. The systematic use of different databases, managed throughout the built knowledge graph, gives new insights toward a better understanding of oncology medicine. Indeed, we reduce some useful medical tasks to well-known problems in theoretical computer science for which efficient algorithms exist.

cs.AI↗

arXiv2024

On Generalizations of Pairwise Compatibility Graphs

A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.

math.CO↗

arXiv2022

(Eternal) Vertex Cover Number of Infinite and Finite Grid Graphs

In the eternal vertex cover problem, mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on its edges by moving to neighbor vertices. The eternal vertex cover problem consists in determining the minimum number of necessary guards. Motivated by previous literature, in this paper, we study the vertex cover and eternal vertex cover problems on regular grids, when passing from infinite to finite version of the same graphs, and we provide either coinciding or very tight lower and upper bounds on the number of necessary guards. To this aim, we generalize the notions of minimum vertex covers and minimum eternal vertex cover in order to be well defined for infinite grids.

cs.DM↗

arXiv2022

On the domination number of $t$-constrained de Bruijn graphs

Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.

math.CO↗

arXiv2022

Multi-Depot Multi-Trip Vehicle Routing with Total Completion Time Minimization

Unmanned aerial vehicles (UAVs) are aircraft whose flights can be fully autonomous without any provision for human intervention. One of the most useful and promising domains where UAVs can be employed is natural disaster management. In this paper, we focus on an emergency scenario and propose the use of a fleet of UAVs that help rescue teams to individuate people needing help inside an affected area. We model this situation as an original graph theoretical problem called Multi-Depot Multi-Trip Vehicle Routing Problem with Total Completion Times minimization (MDMT-VRP-TCT); we go through some problems already studied in the literature that appear somehow similar to it and highlight the differences, propose a mathematical formulation for our problem as a MILP, design a matheuristic framework to quickly solve large instances, and experimentally test its performance. Beyond the proposed application, our solution works in any case in which a multi-depot multi-trip vehicle routing problem must be solved.

math.OC↗

arXiv2017

Visualizing Co-Phylogenetic Reconciliations

We introduce a hybrid metaphor for the visualization of the reconciliations of co-phylogenetic trees, that are mappings among the nodes of two trees. The typical application is the visualization of the co-evolution of hosts and parasites in biology. Our strategy combines a space-filling and a node-link approach. Differently from traditional methods, it guarantees an unambiguous and `downward' representation whenever the reconciliation is time-consistent (i.e., meaningful). We address the problem of the minimization of the number of crossings in the representation, by giving a characterization of planar instances and by establishing the complexity of the problem. Finally, we propose heuristics for computing representations with few crossings.

cs.DS↗

arXiv2017

Some classes of graphs that are not PCGs

A graph $G=(V,E)$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$, $d_{min} \leq d_{max}$, such that each node $u \in V$ is uniquely associated to a leaf of $T$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_{T} (u, v) \leq d_{max}$, where $d_{T} (u, v)$ is the sum of the weights of the edges on the unique path $P_{T}(u,v)$ from $u$ to $v$ in $T$. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. As an example, we use this technique to show that wheels and graphs obtained as strong product between a cycle and $P_2$ are not PCGs.

cs.DM↗

arXiv2015

On Pairwise Compatibility of Some Graph (Super)Classes

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v) <= D' where d_T(u, v) is the sum of weights of the edges on the unique path from `u' to `v' in T. The main issue on these graphs consists in characterizing them. In this note we prove the inclusion in the PCG class of threshold tolerance graphs and the non-inclusion of a number of intersection graphs, such as disk and grid intersection graphs, circular arc and tolerance graphs. The non-inclusion of some superclasses (trapezoid, permutation and rectangle intersection graphs) follows.

cs.DM↗

arXiv2012

All graphs with at most seven vertices are Pairwise Compatibility Graphs

A graph $G$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $l_u$ of $T$ corresponds to a vertex $u \in V$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_{T,w} (l_u, l_v) \leq d_{max}$ where $d_{T,w} (l_u, l_v)$ is the sum of the weights of the edges on the unique path from $l_u$ to $l_v$ in $T$. In this note, we show that all the graphs with at most seven vertices are PCGs. In particular all these graphs except for the wheel on 7 vertices $W_7$ are PCGs of a particular structure of a tree: a centipede.

cs.DM↗

arXiv2011

On relaxing the constraints in pairwise compatibility graphs

A graph $G$ is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $l_u$ of $T$ corresponds to a vertex $u \in V$ and there is an edge $(u,v) \in E$ if and only if $d_{min} \leq d_T (l_u, l_v) \leq d_{max}$ where $d_T (l_u, l_v)$ is the sum of the weights of the edges on the unique path from $l_u$ to $l_v$ in $T$. In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where $\dmin=0$ (LPG) and $\dmax=+\infty$ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.

cs.DM↗

arXiv2011

Sensor Activation and Radius Adaptation (SARA) in Heterogeneous Sensor Networks

In this paper we address the problem of prolonging the lifetime of wireless sensor networks (WSNs) deployed to monitor an area of interest. In this scenario, a helpful approach is to reduce coverage redundancy and therefore the energy expenditure due to coverage. We introduce the first algorithm which reduces coverage redundancy by means of Sensor Activation and sensing Radius Adaptation (SARA)in a general applicative scenario with two classes of devices: sensors that can adapt their sensing range (adjustable sensors) and sensors that cannot (fixed sensors). In particular, SARA activates only a subset of all the available sensors and reduces the sensing range of the adjustable sensors that have been activated. In doing so, SARA also takes possible heterogeneous coverage capabilities of sensors belonging to the same class into account. It specifically addresses device heterogeneity by modeling the coverage problem in the Laguerre geometry through Voronoi-Laguerre diagrams. SARA executes quickly and is guaranteed to terminate. It provides a configuration of the active set of sensors that lifetime and coverage requirements of demanding WSN applications, not met by current solutions. By means of extensive simulations we show that SARA achieves a network lifetime that is significantly superior to that obtained by previous algorithms in all the considered scenarios.

cs.NI↗

arXiv2009

The L(2, 1)-Labeling Problem on Oriented Regular Grids

The L(2, 1)-labeling of a digraph G is a function f from the node set of $G$ to the set of all nonnegative integers such that $|f(x)-f(y)| \geq 2$ if $x$ and $y$ are at distance 1, and $f(x)=f(y)$ if $x$ and $y$ are at distance 2, where the distance from vertex $x$ to vertex $y$ is the length of a shortest dipath from $x$ to $y$. The minimum of the maximum used label over all $L(2, 1)$-labelings of $G$ is called $λ(G)$. In this paper we study the L(2, 1)-labeling problem on squared, triangular and hexagonal grids and for them we compute the exact values of $λ$.

cs.DM↗

arXiv2008

Minimum-energy broadcast in random-grid ad-hoc networks: approximation and distributed algorithms

The Min Energy broadcast problem consists in assigning transmission ranges to the nodes of an ad-hoc network in order to guarantee a directed spanning tree from a given source node and, at the same time, to minimize the energy consumption (i.e. the energy cost) yielded by the range assignment. Min energy broadcast is known to be NP-hard. We consider random-grid networks where nodes are chosen independently at random from the $n$ points of a $\sqrt n \times \sqrt n$ square grid in the plane. The probability of the existence of a node at a given point of the grid does depend on that point, that is, the probability distribution can be non-uniform. By using information-theoretic arguments, we prove a lower bound $(1-ε) \frac nπ$ on the energy cost of any feasible solution for this problem. Then, we provide an efficient solution of energy cost not larger than $1.1204 \frac nπ$. Finally, we present a fully-distributed protocol that constructs a broadcast range assignment of energy cost not larger than $8n$,thus still yielding constant approximation. The energy load is well balanced and, at the same time, the work complexity (i.e. the energy due to all message transmissions of the protocol) is asymptotically optimal. The completion time of the protocol is only an $O(\log n)$ factor slower than the optimum. The approximation quality of our distributed solution is also experimentally evaluated. All bounds hold with probability at least $1-1/n^{Θ(1)}$.

cs.DS↗