Star Convex Research Articles

On maximal Roman domination in graphs: complexity and algorithms

For a simple undirected connected graph G = (V, E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0, 1, 2} with the following properties: (i) For every vertex v ∈ {v ∈ V|f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V|f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V|f(v) ≠ 0} such that N(v) ∩ {u ∈ V|f(u) = 0} ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) = ∑v∈V (G) f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

Further Results on the [k]-Roman Domination in Graphs

In 2016, Beeler et al. defined the double Roman domination as a variation of Roman domination. Sometime later, in 2021, Ahangar et al. introduced the concept of [k]-Roman domination in graphs and settled some results on the triple Roman domination case. In 2022, Amjadi et al. studied the quadruple version of this Roman-domination-type problem. Given any labeling of the vertices of a graph, AN(v) stands for the set of neighbors of a vertex v having a positive label. In this paper we continue the study of the [k]-Roman domination functions ([k]-RDF) in graphs which coincides with the previous versions when 2≤k≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le k \\le 4$$\\end{document}. Namely, f is a [k]-RDF if f(N[v])≥k+|AN(v)|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(N[v])\\ge k+|AN(v)|$$\\end{document} for all v. We prove that the associate decision problem is NP-complete even when restricted to star convex and comb convex bipartite graphs and we also give sharp bounds and exact values for several classes of graphs.

On the complexity of co-secure dominating set problem

A set D⊆V of a graph G=(V,E) is a dominating set of G if every vertex v∈V∖D is adjacent to at least one vertex in D. A set S⊆V is a co-secure dominating set (CSDS) of a graph G if S is a dominating set of G and for each vertex u∈S there exists a vertex v∈V∖S such that uv∈E and (S∖{u})∪{v} is a dominating set of G. The minimum cardinality of a co-secure dominating set of G is the co-secure domination number and it is denoted by γcs(G). Given a graph G=(V,E), the minimum co-secure dominating set problem (Min Co-secure Dom) is to find a co-secure dominating set of minimum cardinality. In this paper, we strengthen the inapproximability result of Min Co-secure Dom for general graphs by showing that this problem can not be approximated within a factor of (1−ε)ln⁡|V| for perfect elimination bipartite graphs and star convex bipartite graphs unless P=NP. On the positive side, we show that Min Co-secure Dom can be approximated within a factor of O(ln⁡|V|) for any graph G with δ(G)≥2. For 3-regular and 4-regular graphs, we show that Min Co-secure Dom is approximable within a factor of 83 and 103, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for 3-regular graphs.

Total 2-Rainbow Domination in Graphs: Complexity and Algorithms

For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.

Scaled boundary isogeometric analysis with C1 coupling for Kirchhoff‐Love theory

AbstractScaled boundary isogeometric analysis (SB‐IGA) describes the computational domain by proper boundary NURBS together with a well‐defined scaling center; see [5]. More precisely, we consider star convex domains whose domain boundaries correspond to a sequence of NURBS curves and the interior is determined by a scaling of the boundary segments with respect to a chosen scaling center. However, providing a decomposition into star shaped blocks one can utilize SB‐IGA also for more general shapes. Even though several geometries can be described by a single patch, in applications frequently there appear multipatch structures. Whereas a C0 continuous patch coupling can be achieved relatively easily, the situation becomes more complicated if higher regularity is required. Consequently, a suitable coupling method is inevitably needed for analyses that require global C1 continuity.In this contribution we apply the concept of analysis‐suitable G1 parametrizations [2] to the framework of SB‐IGA for the C1 coupling of planar domains with a special consideration of the scaling center. We obtain globally C1 regular basis functions and this enables us to handle problems such as the Kirchhoff‐Love plate and shell, where smooth coupling is an issue. Furthermore, the boundary representation within SB‐IGA makes the method suitable for the concept of trimming. In particular, we see the possibility to extend the coupling procedure to study trimmed plates and shells.The approach was implemented using the GeoPDEs package [1] and its performance was tested on several numerical examples. Finally, we discuss the advantages and disadvantages of the proposed method and outline future perspectives.

An effective approach for the nuclei segmentation from breast histopathological images using star-convex polygon

Early detection and patient-driven precise treatment can drastically reduce the elevation in the mortality rate due to breast tumours in women. Therefore, it is inevitable to develop an interpretable model, which detects and segments cancer cell nuclei. The main challenge to achieve this goal is the separation of the individual cell nucleus from the clustered overlapping nuclei without losing the pixel-level shape correspondence of each nucleus. The performance of the most common image processing conventional methods alone has not yielded the expected result for nuclei segmentation from the digitized tissue images either due to over-segmentation or under-segmentation. Here, this issue is addressed by developing an automatic nuclei segmentation algorithm which uses a deep learning-based approach. This paper introduces pixel-level analysis using the star convex polygon approach for segmenting the cell nuclei from the histopathological images. Analysis shows that this technique exemplifies a more accurate segmentation result with a validation IOU score of 87.56%. The true positive rate and the detection of pixel-level shape correspondence are also comparatively high for this method during the test phase.

Algorithmic aspects of Roman {3}-domination in graphs

Let G be a simple, undirected graph. A function g : V(G) → {0, 1, 2, 3} having the property that ∑v∈NG(u)g(v)≥3, if g(u) = 0, and ∑v∈NG(u)g(v)≥2, if g(u) = 1 for any vertex u ∈ G, where NG(u) is the set of vertices adjacent to u in G, is called a Roman {3}-dominating function (R3DF) of G. The weight of a R3DF g is the sum g(V)=∑v∈Vg(v). The minimum weight of a R3DF is called the Roman {3}-domination number and is denoted by γ{R3}(G). Given a graph G and a positive integer k, the Roman {3}-domination problem (R3DP) is to check whether G has a R3DF of weight at most k. In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. The minimum Roman {3}-domination problem (MR3DP) is to find a R3DF of minimum weight in the input graph. We show that MR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. We propose a 3(1 + ln(Δ − 1))-approximation algorithm for the MR3DP, where Δ is the maximum degree of G and show that the MR3DP problem cannot be approximated within (1 − ε)ln|V| for any ε > 0 unless NP ⊆ DTIME(|V|O(loglog|V|)). Next, we show that the MR3DP problem is APX-complete for graphs with maximum degree 4. We also show that the domination and Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, an ILP formulation for MR3DP is proposed.

Algorithmic Aspects of Quasi-Total Roman Domination in Graphs

For a simple, undirected, connected graph G=(V,E), a function f : V(G) →{0, 1, 2} which satisfies the following conditions is called a quasi-total Roman dominating function (QTRDF) of G with weight f(V(G))=ΣvEV(G) f(v).C1). Every vertex ueV for which f(u) = 0 must be adjacent to at least one vertex v with f(v) = 2, and C2). Every vertex ueV for which f(u) = 2 must be adjacent to at least one vertex v with f(v)≥1. For a graph G, the smallest possible weight of a QTRDF of G denoted γqtR(G) is known as the quasi-total Roman domination number of G. The problem of determining γqtR(G) of a graph G is called minimum quasi-total Roman domination problem (MQTRDP). In this paper, we show that the problem of determining whether G has a QTRDF of weight at most l is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. On the positive side, we show that MQTRDP for threshold graphs, chain graphs and bounded treewidth graphs is linear time solvable. Finally, an integer linear programming formulation for MQTRDP is presented.

Total vertex-edge domination in graphs: Complexity and algorithms

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.

Vertex-Edge Roman Domination in Graphs: Complexity and Algorithms

For a simple, undirected graph $G(V,E)$, a function $h : V(G) rightarrow lbrace 0, 1, 2rbrace$ such that each edge $ (u,v)$ of $G$ is either incident with a vertex with weight at least one or there exists a vertex $w$ such that either $(u,w) in E(G)$ or $(v,w) in E(G)$ and $h(w) = 2$, is called a vertex-edge Roman dominating function (ve-RDF) of $G$. For a graph $G$, the smallest possible weight of a ve-RDF of $G$ which is denoted by $gamma_{veR}(G)$, is known as the textit{vertex-edge Roman domination number} of $G$. The problem of determining $gamma_{veR}(G)$ of a graph $G$ is called minimum vertex-edge Roman domination problem (MVERDP). In this article, we show that the problem of deciding if $G$ has a ve-RDF of weight at most $l$ for star convex bipartite graphs, comb convex bipartite graphs, chordal graphs and planar graphs is NP-complete. On the positive side, we show that MVERDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, a 2-approximation algorithm for MVERDP is presented. It is also shown that vertex cover and vertex-edge Roman domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MVERDP is presented.

Algorithmic aspects of 2-secure domination in graphs

Let G(V, E) be a simple, connected and undirected graph. A dominating set $$S \subseteq V$$ is called a 2-secure dominating set (2-SDS) in G, if for each pair of distinct vertices $$v_1,v_2 \in V$$ there exists a pair of distinct vertices $$u_1,u_2 \in S$$ such that $$u_1 \in N[v_1]$$ , $$u_2 \in N[v_2]$$ and $$(S {\setminus } \{u_1,u_2\}) \cup \{v_1,v_2 \}$$ is a dominating set in G. The size of a minimum 2-SDS in G is said to be 2-secure domination number denoted by $$\gamma _{2s}(G)$$ . The 2-SDM problem is to check if an input graph G has a 2-SDS S, with $$ \vert S \vert \le k$$ , where $$ k \in \mathbb {Z}^+ $$ . It is proved that for bipartite graphs 2-SDM is NP-complete. In this paper, we prove that the 2-SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs. We reinforce the existing NP-complete result for bipartite graphs, by proving 2-SDM is NP-complete for some subclasses of bipartite graphs specifically, comb convex bipartite and star convex bipartite graphs. We prove that this problem is linear time solvable for bounded tree-width graphs. We also show that the 2-SDM is W[2]-hard even for split graphs. The M2SDS problem is to find a 2-SDS of minimum size in the given graph. We give a $$ \varDelta +1 $$ -approximation algorithm for M2SDS, where $$ \varDelta $$ is the maximum degree of the given graph and prove that M2SDS cannot be approximated within $$ (1 - \epsilon ) \ln (\vert V \vert ) $$ for any $$ \epsilon > 0 $$ unless $$ NP \subseteq DTIME(\vert V \vert ^{ O(\log \log \vert V \vert )}) $$ . Finally, we prove that the M2SDS is APX-complete for graphs with $$\varDelta =4.$$

Complexity issues of perfect secure domination in graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

Algorithmic aspects of secure connected domination in graphs

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \} $ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $ \gamma_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $ G$ and a positive integer $ k,$ the Secure Connected Domination (SCDM) problem is to check whether $ G $ has a secure connected dominating set of size at most $ k.$ In this paper, we prove that the SCDM problem is NP-complete for doubly chordal graphs, a subclass of chordal graphs. We investigate the complexity of this problem for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a secure connected dominating set of minimum size in the input graph. We propose a $ (\Delta(G)+1) $ - approximation algorithm for MSCDS, where $ \Delta(G) $ is the maximum degree of the input graph $ G $ and prove that MSCDS cannot be approximated within $ (1 -\epsilon) ln(| V |)$ for any $ \epsilon > 0 $ unless $ NP \subseteq DTIME(| V |^{O(log log | V |)})$ even for bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs with $\Delta(G)=4$.

Complexity Aspects of Variants of Independent Roman Domination in Graphs

For a simple, undirected graph $$G = (V,$$ E), an independent Roman dominating function (IRDF) $$f : V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that every vertex $$v \in V$$ with $$f(v) = 0$$ is adjacent to at least one vertex u, with $$f(u) = 2$$ , and no two vertices assigned positive values are adjacent. An independent Roman $$\lbrace 2 \rbrace $$ -dominating function (IR2DF) is an IRDF f with an additional property that, for every vertex $$v \in V$$ with $$f(v) = 0$$ , there exists at least two vertices $$x, y \in N_G(v)$$ with $$f(x) = f(y) = 1$$ . An independent double Roman dominating function (IDRDF) on G is a function $$f : V \rightarrow \lbrace 0, 1, 2, 3 \rbrace $$ , such that for every vertex $$v \in V$$ , if $$f(v) = 0$$ , then v has at least two neighbors x, y with $$f(x) = f(y) = 2$$ or one neighbor w with $$f(w) = 3$$ , and if $$f(v) = 1$$ , then v must have at least one neighbor w with $$f(w) \ge 2$$ , and no two vertices assigned positive values are adjacent. The weight of an IRDF (IR2DF, IDRDF) f is the sum $$f(V) = \sum _{v \in V}f(v)$$ . Given a graph G and a positive integer k, the independent Roman domination problem (IRDP), independent Roman $$\lbrace 2 \rbrace $$ -domination problem (IR2DP) and independent double Roman domination problem (IDRDP), respectively, is to check whether G has an IRDF, IR2DF, and IDRDF of weight at most k. The IRDP, IR2DP and IDRDP are known to be NP-complete for bipartite graphs. We investigate the complexity of these problems for dually chordal graphs and some subclasses of bipartite graphs, namely, star convex bipartite graphs and comb convex bipartite graphs. We also investigate the complexity of IR2DP and IDRDP for chordal graphs. The minimum independent Roman domination problem (MIRDP), minimum independent Roman $$\lbrace 2 \rbrace $$ -domination problem (MIR2DP), and minimum independent double Roman domination problem (MIDRDP), respectively, are to find an IRDF, IR2DF, and IDRDF of minimum weight in the input graph. We show that MIRDP, MIR2DP, and MIDRDP are linear time solvable for bounded tree-width graphs, chain graphs, and threshold graphs, a subclass of split graphs. We also show that the MIRDP, MIR2DP, and MIDRDP are APX-hard for graphs with bounded degree 4. Finally, we show that the domination problem and IRDP (IR2DP, IDRDP) are not equivalent in computational complexity aspects.

When Is a Helix Stable?

We determine which helical equilibria of an isotropic Kirchhoff elastic rod with clamped ends are stable and which are unstable. Although the set of all helical equilibria is parametrized by four variables, with an additional fifth parameter determined by the rod's material, we show that only three of these five parameters are needed to distinguish between stable and unstable equilibria. We also show that the closure of the set of stable equilibria is star convex. With these results, we are able to compute and visualize the boundary between stable and unstable helices for the first time.

Krasnosel’skii type compression-expansion fixed point theorem for set contractions and star convex sets

In this paper, we give or improve compression-expansion results for set contractions in conical domains determined by balls or star convex sets. In the compression case, we use Potter’s idea of proof, while the expansion case is reduced to the compression one by means of a change of variable. Finally, to illustrate the theory, we give an application to the initial value problem for a system of implicit first order differential equations.

Complexity of Roman {2}-domination and the double Roman domination in graphs

For a simple, undirected graph a Roman {2}-dominating function (R2DF) has the property that for every vertex with f(v) = 0, either there exists a vertex with f(u) = 2, or at least two vertices with The weight of an R2DF is the sum The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by A double Roman dominating function (DRDF) on G is a function such that for every vertex if f(v) = 0, then v has at least two neighbors with or one neighbor w with f(w) = 3, and if f(v) = 1, then v must have at least one neighbor w with The weight of a DRDF is the value The minimum weight of a DRDF is called the double Roman domination number and is denoted by Given an graph G and a positive integer k, the R2DP (DRDP) problem is to check whether G has an R2DF (DRDF) of weight at most k. In this article, we first show that the R2DP problem is NP-complete for star convex bipartite graphs, comb convex bipartite graphs and bisplit graphs. We also show that the DRDP problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Next, we show that are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a -approximation algorithm for the minimum Roman {2}-domination problem and -approximation algorithm for the minimum double Roman domination problem, where Δ is the maximum degree of G.

Algorithmic aspects of Roman domination in graphs

For a simple, undirected graph $$G = (V, E)$$ , a Roman dominating function (RDF) $$f{:}V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that, every vertex u with $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) = 2$$ . The weight of a RDF is the sum $$f(V) = \sum _{v \in V}f(v)$$ . The minimum weight of a RDF is called the Roman domination number and is denoted by $$\gamma _{R}(G)$$ . Given a graph G and a positive integer k, the Roman domination problem (RDP) is to check whether G has a RDF of weight at most k. The RDP is known to be NP-complete for bipartite graphs. We strengthen this result by showing that this problem remains NP-complete for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. We show that $$\gamma _{R}(G)$$ is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph. We show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within $$(1 - \epsilon ) \ln |V|$$ for any $$\epsilon > 0$$ unless $$NP \subseteq DTIME(|V|^{O(\log \log |V|)})$$ and also propose a $$2(1+\ln (\varDelta +1))$$ -approximation algorithm for the MRDP, where $$\varDelta $$ is the maximum degree of G. Finally, we show that the MRDP is APX-complete for graphs with maximum degree 5.

Optimum Cuts in Graphs by General Fuzzy Connectedness with Local Band Constraints

The goal of this work is to describe an efficient algorithm for finding a binary segmentation of an image such that the indicated object satisfies a novel high-level prior, called local band, LB, constraint; the returned segmentation is optimal, with respect to an appropriate graph-cut measure, among all segmentations satisfying the given LB constraint. The new algorithm has two stages: expanding the number of edges of a standard edge-weighted graph of an image; applying to this new weighted graph an algorithm known as an oriented image foresting transform, OIFT. In our theoretical investigation, we prove that OIFT algorithm belongs to a class of general fuzzy connectedness algorithms and so has several good theoretical properties, like robustness for seed placement. The extension of the graph constructed in the first stage ensures, as we prove, that the resulted object indeed satisfies the given LB constraint. We also notice that this graph construction is flexible enough to allow combining it with other high-level constraints. Finally, we experimentally demonstrate that the LB constraint gives competitive results as compared to geodesic star convexity, boundary band, and hedgehog shape prior, all implemented within OIFT framework and applied to various scenarios involving natural and medical images.

A dual scaled boundary finite element formulation over arbitrary faceted star convex polyhedra

A novel technique to formulate arbritrary faceted polyhedral elements in three-dimensions is presented. The formulation is applicable for arbitrary faceted polyhedra, provided that a scaling requirement is satisfied and the polyhedron facets are planar. A triangulation process can be applied to non-planar facets to generate an admissible geometry. The formulation adopts two separate scaled boundary coordinate systems with respect to: (i) a scaling centre located within a polyhedron and; (ii) a scaling centre on a polyhedron’s facets. The polyhedron geometry is scaled with respect to both the scaling centres. Polygonal shape functions are derived using the scaled boundary finite element method on the polyhedron facets. The stiffness matrix of a polyhedron is obtained semi-analytically. Numerical integration is required only for the line elements that discretise the polyhedron boundaries. The new formulation passes the patch test. Application of the new formulation in computational solid mechanics is demonstrated using a few numerical benchmarks.