A sequence of positive integers is graphic if it is the degree sequence of some simple graph , and planaric if it is the degree sequence of some simple planar graph . It is known that if , then has a realization by a forest, hence it is trivially planaric. This condition for to be realized by a forest is tight, since clearly must be even, and it is known that if then does not have a forest realization. In this paper, we seek bounds on that guarantee that if is graphic then it is also planaric. We show that this holds true when , where is the number of 1's in . Conversely, we show that there are graphic sequences with that are non-planaric. For the case , we show that is planaric when . Conversely, we show that there is a graphic sequence with that is non-planaric. In fact, when can be realized by a graph with a 2-page book embedding.

We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then there is a complete characterization that was established more than 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open. We investigate the role of an important class of special partitions, called High-Low partitions, which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analyzing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.

The Degree Realization problem with respect to a graph family $\mathcal{F}$ is defined as follows. The input is a sequence $d$ of $n$ positive integers, and the goal is to decide whether there exists a graph $G \in \mathcal{F}$ whose degrees correspond to $d$. The main challenges are to provide a precise characterization of all the sequences that admit a realization in $\mathcal{F}$ and to design efficient algorithms that construct one of the possible realizations, if one exists. This paper studies the problem of realizing degree sequences by bipartite cactus graphs (where the input is given as a single sequence, without the bi-partition). A characterization of the sequences that have a cactus realization is already known [Rao 1981]. In this paper, we provide a systematic way to obtain such a characterization, accompanied by a realization algorithm. This allows us to derive a characterization for bipartite cactus graphs, and as a byproduct, for several other interesting sub-families of cactus graphs, including bridge-less cactus graphs and core cactus graphs, as well as for the bipartite sub-families of these families. Finally, we also provide a characterization of forcibly bipartite cactus sequences.

The Degree Realization problem requires, given a sequence d of n positive integers, to decide whether there exists a graph whose degrees correspond to d, and to construct such a graph if it exists. A more challenging variant of the problem arises when d has many different realizations, and some of them may be more desirable than others. We study optimized realization problems in which the goal is to compute a realization that optimizes some quality measure. Efficient algorithms are known for the problems of finding a realization with the maximum clique, the maximum independent set, or the minimum vertex cover. In this paper, we focus on two problems for which such algorithms were not known. The first is the Degree Realization with Minimum Dominating Set problem, where the goal is to find a realization whose minimum dominating set is minimized among all the realizations of the given sequence d. The second is the Degree Realization with Maximum Matching problem, where the goal is to find a realization with the largest matching among all the realizations of d. We present polynomial time realization algorithms for these two open problems. A related problem of interest and importance is characterizing the sequences with a given value of the optimized function. This leads to an efficient computation of the optimized value without providing the realization that achieves that value. For the Maximum Matching problem, a succinct characterization of degree sequences with a maximum matching of a given size was known. This paper provides a succinct characterization of sequences with minimum dominating set of a given size.

Assume we inspect a specimen represented as a collection of points and our task is to learn a physical value associated with each point. However, performing a direct measurement is impossible since it damages the specimen. The alternative is to employ aggregate measuring techniques (e.g., CT or MRI), whereby measurements are taken over subsets of points, and the exact values at each point are subsequently extracted by computational methods. In the MINIMUM SURGICAL PROBING problem (MSP) the inspected specimen is represented by a graph G and a vector l is an element of R-n that assigns a value l(i) to each vertex i. An aggregate measurement (called probe) centred at vertex i captures its entire neighbourhood, i.e., the outcome of a probe at i is P-i = Sigma(j is an element of N(i)boolean OR{i})l(j) where N(i) is the open neighbourhood of vertex i. Bar-Noy et al. (2022) gave a criterion whether the vector l can be recovered from the collection of probes P = { P-v | v is an element of V(G)} alone. However, there are graphs where the vector l cannot be recovered from P alone. In these cases, we are allowed to use surgical probes. A surgical probe at vertex i returns l(i). The objective of MSP is to recover l from P and G using as few surgical probes as possible. In this paper, we introduce the WEIGHTED MINIMUM SURGICAL PROBING (WMSP) problem in which a vertex i may have an aggregation coefficient w(i), namely P-i = Sigma(j is an element of N(i)) l(j) + w(i) l(i). We show that WMSP can be solved in polynomial time. Moreover, we analyse the number of required surgical probes depending on the weight vector w. For any graph, we give two boundaries outside of which no surgical probes are needed to recover the vector l. The boundaries are connected to the (Signless) Laplacian matrix.In addition, we consider the special case where w = (0) over right arrow and explore the range of possible behaviour of WMSP by determining the number of surgical probes necessary in certain graph families, such as trees and various grid graphs. (c) 2023 Elsevier B.V. All rights reserved.

We study the question of whether a sequence d = (d_1,d_2, … , d_n) of positive integers is the degree sequence of some outerplanar (a.k.a. 1-page book embeddable) graph G. If so, G is an outerplanar realization of d and d is an outerplanaric sequence. The case where ∑ d ≤ 2n - 2 is easy, as d has a realization by a forest (which is trivially an outerplanar graph). In this paper, we consider the family 𝒟 of all sequences d of even sum 2n≤∑ d ≤ 4n-6-2ω _1 , where ω _x is the number of x’s in d. (The second inequality is a necessary condition for a sequence d with ∑ d≥ 2n to be outerplanaric.) We partition 𝒟 into two disjoint subfamilies, 𝒟=𝒟_NOP∪𝒟_2PBE , such that every sequence in 𝒟_NOP is provably non-outerplanaric, and every sequence in 𝒟_2PBE is given a realizing graph G enjoying a 2-page book embedding (and moreover, one of the pages is also bipartite).

The paper concerns the question of which properties of a graph are already determined by its degree sequence. The classic degree realization problem asks to characterize graphic sequences, i.e., sequences of positive integers, which are the degree sequence of some simple graph. Erdos and Gallai [11] solved this problem. Havel and Hakimi [14,17] provide a different characterization implying an algorithm to generate a realizing graph (if one exists). It is known that a graphic sequence can have several non-isomorphic realizations. We characterize graphic sequences where every realization has some given graph property P. Such sequences are called forcibly P-graphic. In particular, we present complete results characterizing forcibly (connected) bipartite, forcibly acyclic, and forcibly tree-graphic sequences. In those four models, we also characterize the sequences with a unique realizing graph, called unigraphic sequences. Finally, we address the problem of counting the number of sequences in each model.(c) 2023 Elsevier B.V. All rights reserved.

The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing degree sequences that can be realized by a bipartite (simple) graph has two variants. In the simpler one, termed BDR $$^P$$ , the partition of the degree sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are again two variants. For BDR $$^P$$ , where the partition is given as part of the input, a complete characterization was known for determining whether the bi-sequence is r-max-bigraphic, namely, if there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most r. We present a complete characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most t. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence.

This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d1,…,dn) of non-negative integers, the goal is to construct a simple graph with vertices v1,…,vn such that for every i∈[1,n], the maximum degree in the neighborhood of vi is exactly di (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.

It is often of interest to sample vertices from a graph with a bias towards higher-degree vertices. One well-known method, which we call random neighbor or RN, involves taking a vertex at random and exchanging it for one of its neighbors. Loosely inspired by the friendship paradox, the method is predicated on the fact that the expected degree of the neighbor is greater than or equal to the expected degree of the initial vertex. Another method that is actually perfectly analogous to the friendship paradox is random edge, or RE, where an edge is sampled at random, and then one of the two endpoint vertices is selected at random. Obviously, random sampling is only required when full knowledge of the graph is unattainable. But, while it is true in most cases that knowledge of all vertices’ degrees cannot be obtained, it is often trivial to learn the degree of specific vertices that have already been isolated. In light of this, we suggest a tweak to both RN and RE, inclusive random sampling. In inclusive random neighbor (IRN) the initial vertex and the selected neighbor are considered, in inclusive random edge (IRE) the two endpoint vertices are, and in both cases, we learn the degree of each and select the vertex of higher degree. This paper explores inclusive random sampling through theoretical analysis and experimentation. We establish meaningful bounds on IRN and IRE’s performances, in particular in comparison to each other and to their exclusive counterparts. Our analyses highlight differences of the original, exclusive versions as well. The results provide practical insight for strategizing a random sampling method, and also highlight graph characteristics that impact the question of which methods will perform strongly in which graphs.

Network realization problems require, given a specification π for some network parameter (such as degrees, distances or connectivity), to construct a network G conforming to π , or to determine that no such network exists. In this paper we study composed profile realization, where the given instance consists of two or more profile specifications that need to be realized simultaneously. To gain some understanding of the problem, we focus on two classical profile types, namely, degrees and distances , which were (separately) studied extensively in the past. We investigate a wide spectrum of variants of the composed distance and degree realization problem. For each variant we either give a polynomial-time realization algorithm or establish NP hardness. In particular: We consider both precise specifications and range specifications, which specify a range of permissible values for each entry of the profile. We consider realizations by both weighted and unweighted graphs. We also study settings where the realizing graph is restricted to specific graph classes, including trees and bipartite graphs.

We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then a complete characterization has been established over 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open. We investigate the role of an important class of special partitions, called High-Low partitions , which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analysing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.

Random neighbor sampling, or RN, is a method for sampling vertices with an average degree greater than the mean degree of the graph. It samples a vertex, then exchanges it for one of its neighbors which is assumed to be of higher degree. While considerable research has analyzed various aspects of RN, the extra cost of sampling a second vertex is typically not addressed. In this paper, we offer an analysis of RN from the perspective of cost. We define three separate costs, the cost of sampling a vertex, the cost of sampling a neighbor, and the cost of selecting a vertex (as opposed to discarding it in exchange for another). We introduce variations to RN, RVN which retains the first sampled vertex, and RN-RV and RVN-RV which are 'hybrid' methods that sample differently in two separate phases. We study all of these methods in terms of the costs we define. The cost-benefit analysis highlights the methods' strengths and weaknesses, with a specific focus on their respective performances for finding high-degree vs. low-degree vertices. The analyses and results provide a far richer understanding of RN and open a new area of exploration for researching additional sampling methods that are best suited for cost-effectiveness in light of particular goals.

Consider the following natural variation of the degree realization problem. Let G=(V, E) be a simple undirected graph of order n. Let f ∈ℝ_≥ 0^n be a vector of vertex requirements, and let w∈ℝ_≥ 0^n be a vector of provided services at the vertices. Then w satisfies f on G if the constraints ∑ _j ∈ N(i) w_j = f_i are satisfied for all i ∈ V , where N(i) denotes the neighborhood of i. Given a requirements vector f, the Weighted Graph Realization problem asks for a suitable graph G and a vector w of provided services that satisfy f on G. In [7] it is observed that any requirement vector where n is even can be realized. If n is odd, the problem becomes much harder. For the unsolved cases, the decision of whether f is realizable or not can be formulated as whether f_n (the largest requirement) lies within certain intervals. In [5] some intervals are identified where f can be realized, and their complements form n-3/2 connected intervals (“unknown domains”) which we give odd indices k = 1,3,… , n-4 . The unknown domain for k=1 is shown to be unrealizable. Our main result presents structural properties that a graph must have if it realizes a vector in one of these unknown domains for k ≥ 3 . The unknown domains are characterized by inequalities which we translate to graph properties. Our analysis identifies several realizable sub-intervals, and shows that each of the unknown domains has at least one sub-interval that cannot be realized.

This paper addresses the classical problem of characterizing degree sequences that can be realized by a bipartite graph. For the simpler variant of the problem, where a partition of the sequence into the two sides of the bipartite graph is given as part of the input, a complete characterization was given by Gale and Ryser over 60 years ago. However, the general question, in which both the partition and the realizing graph need to be determined, is still open. This paper provides an overview of some of the known results on this problem in interesting special cases, including realizations by bipartite graphs and bipartite multigraphs.

Random neighbor sampling, or RN, is a method for sampling vertices with a mean degree greater than that of the graph. Instead of naïvely sampling a vertex from a graph and retaining it (‘random vertex’ or RV), a neighbor of the vertex is selected instead. While considerable research has analyzed various aspects of RN, the extra cost of sampling a second vertex is typically not addressed. This paper explores RN sampling from the perspective of cost. We break down the cost of sampling into two distinct costs, that of sampling a vertex and that of sampling a neighbor of an already sampled vertex, and we also include the cost of actually selecting a vertex/neighbor and retaining it for use rather than discarding it. With these three costs as our cost-model, we explore RN and compare it to RV in a more fair manner than comparisons that have been made in previous research. As we delve into costs, a number of variants to RN are introduced. These variants improve on the cost-effectiveness of RN in regard to particular costs and priorities. Our full cost-benefit analysis highlights strengths and weaknesses of the methods. We particularly focus on how our methods perform for sampling high-degree and low-degree vertices, which further enriches the understanding of the methods and how they can be practically applied. We also suggest ‘two-phase’ methods that specifically seek to cover both high-degree and low-degree vertices in separate sampling phases.

Consider the following natural variation of the degree realization problem. Let G = ( V , E ) be a simple undirected graph of order n . Let f ∈ R ≥ 0 n be a vector of vertex requirements , and let w ∈ R ≥ 0 n be a vector of provided services at the vertices. Then w satisfies f on G if the constraints ∑ j ∈ N ( i ) w j = f i are satisfied for all i ∈ V, where N ( i ) denotes the neighbourhood of vector i . Given a requirements vector f , the Vertex-Weighted Graph Realization problem asks for a suitable graph G and a vector w of provided services that satisfy f on G . In this paper, we consider two avenues. We initiate a study that focuses on weighted realizations where the graph is required to be of a specific class by providing a full characterization of realizable requirement vectors for paths and acyclic graphs. However, checking the respective criteria is shown to be NP-hard. In the second part, we advance the study in general graphs which was started in [2] . For the unsolved cases, the question of whether a vector f is realizable can be formulated as whether its largest requirement lies within certain intervals. We describe several new, realizable intervals and show the existence of an interval that cannot be realized. The complete classification for general graphs is an open problem.

This paper presents and studies a generalization of the microscopic image reconstruction problem ( MIR ) introduced by Frosini and Nivat (2007) and Nivat (2002). Consider a specimen for inspection, represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value ℓ x, which we would like to determine. However, it might be that obtaining these values precisely (by what we call a surgical probe ) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as EM, CT, US or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods. In this paper, we extend the MIR framework in a number of ways. First, we consider a generalized setting where the inspected object is represented by an arbitrary graph G, and the vector ℓ ∈ R n assigns a value ℓ v to each node v. A probe centred at a node v will capture a window encompassing its entire neighbourhood N [ v ], i.e., the outcome of a probe centred at v is P v = ∑ w ∈ N [ v ] ℓ w. We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector ℓ from the collection of probes, P = { P v ∣ v ∈ V }. We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P is inconclusive, in the sense that there may be more than one vector ℓ yielding P). Assume that surgical probes are technically available, yet are expensive or risky, and must be used sparingly. We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of ordinary (aggregate) probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP). Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviours of the Minimum Surgical Probing problem by determining the number of surgical probes necessary in certain specific graph families, such as perfect k-ary trees, paths, cycles, grids, tori, tubes and hypercubes.

A network realization problem involves a given specification π for some network parameters (such as vertex degrees or inter-vertex distances), and requires constructing a network G that satisfies π , if possible. In many settings, it may be difficult or impossible to come up with a precise realization (e.g., the specification data might be inaccurate, or the reconstruction problem might be computationally infeasible). In this expository paper, we review various alternative approaches for coping with these difficulties by relaxing the requirements, discuss the resulting problems and illustrate some (precise or approximate) solutions.