Convex Bipartite Graphs 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.

Fair allocation algorithms for indivisible items under structured conflict constraints

We consider the fair allocation of indivisible items to several agents with additional conflict constraints. These are represented by a conflict graph where each item corresponds to a vertex of the graph and edges in the graph represent incompatible pairs of items which should not be allocated to the same agent. This setting combines the issues of Partition and Independent Set and can be seen as a partial coloring of the conflict graph. In the resulting optimization problem, each agent has its own valuation function for the profits of the items. We aim at maximizing the lowest total profit obtained by any of the agents. In a previous paper, this problem was shown to be strongly NP-hard for several well-known graph classes, e.g., bipartite graphs and their line graphs. On the other hand, it was shown that pseudo-polynomial time algorithms exist for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. In this contribution, we extend this line of research by developing pseudo-polynomial time algorithms that solve the problem for the class of convex bipartite conflict graphs, graphs of bounded clique-width, and graphs of bounded tree-independence number. The algorithms are based on dynamic programming and also permit fully polynomial-time approximation schemes (FPTAS).

Fair Allocation of Indivisible Items with Conflict Graphs

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Namely, we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both Partition and Independent Set. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).

Exact square coloring of certain classes of graphs: Complexity and algorithms

A vertex coloring of a graph G=(V,E) is called an exact square coloring of G if any two vertices at distance exactly 2 receive different colors. The minimum number of colors required by an exact square coloring is called the exact square chromatic number of G and is denoted by χ[#2](G). Given a graph G and a positive integer k, the Exact Square Coloring Problem is to decide whether G admits an exact square coloring using k colors. It is known that Exact Square Coloring Problem is NP-complete for chordal graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete for undirected path graphs, which is a proper subclass of chordal graphs. However, we give linear time algorithms for computing the exact square chromatic number of proper interval graphs and threshold graphs, which are proper subclasses of chordal graphs. Moreover, for a proper interval graph G, we show that χ[#2](G)≤3. We also propose a polynomial time algorithm to produce an exact square coloring of a block graph G using at most χ[#2](G)+1 colors. Next, we study a lower bound of χ[#2](G). A subset S of vertices of a graph G=(V,E) is called an exact square clique of G if the distance between any two vertices in S is exactly 2. The cardinality of the maximum exact square clique of G is called the exact square clique number of G and is denoted by ω[#2](G). Clearly, ω[#2](G)≤χ[#2](G). Given a graph G and a positive integer k, the problem of deciding whether ω[#2](G) is at least k, is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we strengthen these results by proving that this problem remains NP-complete for undirected path graphs, perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs. We also compute the exact value of ω[#2](G) for proper interval graphs, threshold graphs, block graphs and convex bipartite graphs.

Steiner tree in k-star caterpillar convex bipartite graphs: a dichotomy

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y is a convex bipartite graph, if there is an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ is consecutive with respect to the ordering of X, and G is said to have convexity with respect to X. A k-star caterpillar is a tree with a collection of stars with each star having k vertices of degree one whose roots are joined by a path. For a bipartite graph with partitions X and Y, we associate a k-star caterpillar on X such that for each vertex in Y, its neighborhood induces a tree. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph $$G=(V,E)$$ and a subset of vertices $$R \subseteq V(G)$$ , the objective is to find a minimum cardinality set $$S \subseteq V(G)$$ such that the set $$R \cup S$$ induces a connected subgraph. In this paper, we present the following dichotomy result: we show that STREE is NP-complete for 1-star caterpillar convex bipartite graphs and polynomial-time solvable for 0-star caterpillar convex bipartite graphs (also known as convex bipartite graphs). We also strengthen the well-known result of Müller and Brandstädt (Theoret Comput Sci 53(2-3):257-265, 1987), which says STREE in chordal bipartite graphs is NP-complete (reduction instances are 3-star caterpillar convex bipartite graphs). As an application, we use our STREE results to solve: (i) the classical dominating set problem in convex bipartite graphs, (ii) STREE on interval graphs, linear time.

The multicolored graph realization problem

We introduce the multicolored graph realization problem (MGR). The input to this problem is a colored graph (G,φ), i.e., a graph G together with a coloring φ on its vertices. We associate each colored graph (G,φ) with a cluster graph (Gφ) in which, after collapsing all vertices with the same color to a node, we remove multiple edges and self-loops. A set of vertices S is multicolored when S has exactly one vertex from each color class. The MGR problem is to decide whether there is a multicolored set S so that, after identifying each vertex in S with its color class, G[S] coincides with Gφ.The MGR problem is related to the well-known class of generalized network problems, most of which are NP-hard, like the generalized Minimum Spanning Tree problem. The MGR is a generalization of the multicolored clique problem, which is known to be W[1]-hard when parameterized by the number of colors. Thus, MGR remains W[1]-hard, when parameterized by the size of the cluster graph. These results imply that the MGR problem is W[1]-hard when parameterized by any graph parameter on Gφ, among which lies treewidth. Consequently, we look at the instances of the problem in which both the number of color classes and the treewidth of Gφ are unbounded. We consider three natural such graph classes: chordal graphs, convex bipartite graphs and 2-dimensional grid graphs. We show that MGR is NP-complete when Gφ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds even for graphs with bounded degree. We provide a complexity dichotomy with respect to cluster size.

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.

Counting dominating sets in some subclasses of bipartite graphs

Counting dominating sets in a graph is a #P-complete problem even for bipartite graphs. This paper concerns the class of tree convex bipartite graphs, which is a subclass of bipartite graphs. A bipartite graph G with bipartition (X,Y) is called tree convex, if a tree T defined on X exists such that for every vertex y in Y, the neighbors of y form a subtree in T. For each y in Y, when the neighbors of y form a path or a rooted directed path in T, the tree convex bipartite graph is called path-tree bipartite or rooted path-tree bipartite, respectively. This work proves that the problem of counting dominating sets remains #P-complete for path-tree bipartite graphs but admits a polynomial-time solution for the more tightly constrained subclass of rooted path-tree bipartite graphs.

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex uin U, the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex uin U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n^2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.

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 $\Delta(G)=4$.

Injective coloring of some subclasses of bipartite graphs and chordal graphs

A vertex coloring of a graph G=(V,E) that uses k colors is called an injectivek-coloring of G if no two vertices having a common neighbor have the same color. The minimum k for which G has an injectivek-coloring is called the injective chromatic number of G. Given a graph G and a positive integer k, the Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. It is known that Decide Injective Coloring Problem is NP-complete for bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs, which are proper subclasses of bipartite graphs. Moreover, we show that for every ϵ>0, it is not possible to efficiently approximate the injective chromatic number of a perfect elimination bipartite graph within a factor of n13−ϵ unless ZPP = NP. On the positive side, we propose a linear time algorithm for biconvex bipartite graphs and O(nm) time algorithm for convex bipartite graphs for finding the optimal injective coloring. We prove that the injective chromatic number of a chordal bipartite graph can be determined in polynomial time. It is known that Decide Injective Coloring Problem is NP-complete for chordal graphs. We give a linear time algorithm for computing the injective chromatic number of proper interval graphs, which is a proper subclass of chordal graphs. Decide Injective Coloring Problem is also known to be NP-complete for split graphs. We show that Decide Injective Coloring Problem remains NP-complete for K1,t-free split graphs for t≥4 and polynomially solvable for t≤3.

ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.